Topological Spaces and Groups : Part 1: Fast Review

Reference:

Topological Transformation Groups by Deane Montgomery and Leo Zippin, Dover Publications, Available in Amazon India.

My motivation:

We present preliminary and somewhat elementary facts of general spaces and groups. Proofs are given in considerable detail and there are examples which may be of help to a reader for whom the subject is new.

(The purpose of this blog is basically improvement in my understanding of the subject. If, by sharing, it helps some student/readers, that would be great. ) (Also, I found that reading/solving just one main text like Topology by James Munkres left me craving for more topological food: I feel I must also know how the founding fathers discovered topology)…

1.0 Introduction:

We use standard set-theoretic symbols : capitals A, B etc. for sets, A \bigcup B for the union of sets (elements in one or both), and A \bigcap B for the intersection (elements in both) etc.

1.1 Spaces (Topological Spaces);

The term space is sometimes used in mathematical literature in a very general sense to denote any collection whose individual objects are called points, but in topology the term space is used only when some further structure is specified for the collection. As the term will be used in this book it has a meaning which is convenient in studying topological groups. The definition is as follows:

DEFINITION: A topological space (or more simply space) is a non-empty set of points certain subsets of which are designated as open and where, moreover, these open sets are subject to the following conditions:

  1. The intersection of any finite number of open sets is open.
  2. The union of any number of open sets is open.
  3. The empty set and the whole space are open.
  4. To each pair of distinct points of space there is associated at least one open set which contains one of the points and does not contain the other.

A space is called discrete if each point is an open set.

Condition (4) is known as the T_{0} separation axiomi in the terminology of Paul Alexandroff and Herr Heinz Hopf. The first three conditions define a topological space in their terminology. The designated system of open sets is the essential part of the topology, and the same set of points can become a topological space in many ways by choosing different systems of subsets designated as open.

1.2. Homeomorphisms

DEFINITION. A homeomorphism is a one-to-one relation between all points of one topological space and all points of a second which puts the open sets of the two spaces in one-one correspondence; the spaces are topologically equivalent.

The notion of homeomorphism is reflexive, symmetric, and transitive so that it is an equivalence relation in a given set of topological spaces.

EXAMPLES of topological spaces:

Let E_{1} denote the set of all real numbers in its customary topology: the open intervals are the sets \{ y: x < y < z\} for every x < z. The open sets are those which are unions of open intervals together with the empty set and the whole space.

Let R_{1} \subset E_{1} denote the set of numbers in the closed interval 0 \leq y \leq 1, where for the moment we take this subset without a topology. This set gives distinct topological spaces as follows:

  1. Topologize R_{1} as customarily : the open sets are the intersections of R_{1} with the open sets of E_{1}.
  2. Topologize R_{1} discretely, that is, let every subset be open.
  3. Topologize R_{1} by the choice : open sets are the null set, the whole space and for each x of R_{1} the set \{ z, x < z \leq 1\}
  4. Topologize R_{1} by the choice: the complement of any finite set is open and the empty set and whole space are open

In the sequel R_{1} will denote the closed unit interval and E_{1} the set of all reals in the customary topology. A set homeomorphic to R_{1} is called an arc. A set homeomorphic to a circle is called a simply closed curve.

1.3 Basis

DEFINITION. A collection \{ Q_{a}\} of open sets of a space is called a basis for open sets if every open set (except possibly the null set) in the space can be represented as a union of sets in \{ Q_{a}\}. It is called a sub-basis if every open set can be represented as a union of finite intersection of sets in \{ Q_{a}\} (except possibly the null set).

A collection \{ Q_{a}\} of open sets of a space S is a basis if and only for every open set Q in S and x \in Q, there is a Q_{a} \in \{ Q_{a} \} such that x \in Q_{a} \subset Q.

If a collection has this property at a particular point x then the collection is called a basis at x.

If a set together with certain subsets are called a sub-basis, then another family of subsets is determined from the sub-basis by taking arbitrary unions and finite intersections. This new family (with the null set added if necessary) then satisfies conditions 1, 2, 3 for open sets of a topological space. Whether 4 will also be satisfied depends on the original family of sets.

EXAMPLE. Let E_{1} denote the space of real numbers in its usual topology. For each pair of rationals r_{1} < r_{2} let \{ r_{1}, r_{2} denote the set of reals r_{1}<x<r_{2}. This countable collection of open sets is a basis.

A space is said to be separable or to satisfy the second countability axiom if it has a countable basis. A space is said to satisfy the “first countability axiom” if it has a countable basis at each point.

EXAMPLE. Let S denote a topological space and let F denote a collection of real valued functions f(x), where x \in S. If f_{0} is a particular element of F then for each positive integer n let Q(f_{0},n)=\{ f \in F: |f(x)-f_{0}(x)| < \frac{1}{n}\} for all x. We may topologize F by choosing the sets Q(f_{0},n) for all f_{0} and n as a sub-basis. The topological space so obtained has a countable basis at each point in many important cases.

1.4 Topology of subsets

Let S be a topological space, T a subset. Let Q \bigcap T be called open in T or open relative to T if Q is open in S. With open sets defined in this way T becomes a topological space and the topology so defined in T is called the induced or relative topology. If S has a countable basis and T \subset S, then T has a countable basis in the induced topology.

DEFINITION. A subset X \subset S is called closed if the complement S - X is open. If X \subset T \subset S then X is called closed in T when T - X is open in T.

Notice that T closed in S and X closed in T implies that X is closed in S. The corresponding assertion for relatively open sets is also true.

It can be seen that finite unions and arbitrary intersections of closed sets are again closed.

DEFINITION. If K \subset S, the intersection of all closed subsets of S which contain K is called the closure of K and is denoted by \overline{K}. If K is closed, K = \overline{K}.

15 Continuous maps

Let S and T be topological spaces and f a map of S into T f: S \rightarrow T, that is, for each x in S, y=f(x) is a point of T. If the inverse of each relatively open set in f(S) is an open set in S, then f is called continuous. In case f(S)=T then f is continuous and V open in T imply f^{-1}(V) is open in S. The map is called an open map if it carries open sets to open sets.

If f is a continuous map of S onto T (that is, f(S)=T) and if f^{-1} is also single valued and continuous, then f and f^{-1} is also single valued and continuous, then f and f^{-1} are homeomorphisms and S and T are homeomorphic or topologically equivalent.

EXAMPLE. The map f(t) = \exp{2\pi it} is a continuous and open map of E_{1} onto a circle (circumference) in the complex plane.

EXAMPLE. Let K denote the cylindrical surface, described in x ,y, z coordinates in three-space by x^{2}+y^{2}=1. Let f_{1} denote the map of K onto E_{1} given by (x,y,z) \rightarrow (0,0,z), let f_{2} the map (x,y,z) \rightarrow (x,y, |z|) of K into K. All three maps are continuous, the first two are open and f_{1} and f_{2} are also closed, that is, they map closed sets into closed sets.

1.6 Topological products

The space of n real variables (x_{1}, x_{2}, \ldots, x_{n}) from -\infty < x_{i} < \infty, where i=1,2,\ldots, n and the cylinder K of the preceding example are instances of topological products.

Let A denote any non-empty set of indices and suppose that to each a \in A there is associated a topological space S_{a}. The totality of functions f defined on A such that f(a) \in S_{a} for each a \in A is called the product of the spaces S_{a}. When topologized as below it will be denoted by PROD S_{n}; we also use the standard symbol \times thus E \times B is the set of ordered pairs (a,b) e \in E, b \in B.

The standard topology for this product space is defined as follows: For each positive integer n, for each choice of n indices a_{1}, a_{2}, \ldots, a_{n} and for each choice of a non empty open set in S_{a_{i}}

U_{a_{i}} \subset S_{a_{i}} for i=1,2, \ldots, n

consider the set of functions f \in PROD \hspace{0.1in} S_{n} for which f(a_{i}) \in U_{a_{i}} for i=1,2, \ldots, n

Let the totality of these sets be a sub-basis for the product. The resulting family of open sets satisfies the definition of space in 1.1

EXAMPLE 1.

The space E_{n}= E_{1} \times E_{1} \times \ldots \times E_{1}, n copies, is the space of n real variables; here A=\{ 1, 2, 3, \ldots, n\} and each S_{i} is homeomorphic to E_{1} (1.2). Let x_{i} \in S_{i}. Then, (x_{1}, x_{2}, \ldots, x_{n}) are the co-ordinates of a point of E_{n}. It can be verified that the sets U_{m}(x) where m \in \mathcal{N}, of points of E_{n}, whose Euclidean distance from x = (x_{1}, x_{2}, \ldots, x_{n}) is less than \frac{1}{m} form a basis at x. The subset R_{1} \times R_{1}\times \ldots \times R_{1} is an n-cell.

EXAMPLE 2.

Let A be of arbitrary cardinal power and let each S_{n}, a \in A, be homemorphic to C_{1}, the circumference of a circle. Then PROD S_{n} is a generalized torus. If A consists of n objects, the product space is the n-dimensional torus. For n=2, we get the torus.

EXAMPLE 3.

Let D=S_{1} \times S_{2} \times \ldots \times S_{n} \times \ldots , where n \in \mathcal{N} where each S_{i} is a pair of points — conveniently regarded as the “same” pair, and designated 0 and 2. This is the Cantor Discontinuum, of Cantor Middle Third set. It is homeomorphic to the subset of the unit interval defined by the convergent series: D: \sum { a_{n} / 3^{n}}, where a_{n}=0, 2. This example will be described in another way in the next section.

THEOREM :

Let F_{a} be a closed subset of the topological space S_{a}, and a \in A. Then PROD F_{n} is a closed subset of PROD S_{a}.

Proof: The reader is requested to try. It is quite elementary.

1.7 Compactness:

DEFINITION: A topological space S is compact if every collection of open sets whose union covers S contains a finite subcollection whose union covers S.

EXAMPLE 1.

The unit interval R_{1} Thus let \{ Q\} denote a collection of open sets covering R_{1}. Let F denote the set of points x \in R such that the interval 0 \leq y \leq x can be covered by a finite subcollection of \{ Q \}. Then F is not empty and is both open and closed. Hence, by the Dedekind cut postulate, or the existence of least upper bounds, or the connectedness of R_{1} it follows that F = R_{1} To illustrate the concept of compactness, consider the open sets W_{n} \subset R_{1}, W_{n}: \frac{1}{3n}< x < \frac{1}{n}, n \in \mathcal{N}. This collection does not cover R_{1}. Let W_{n} be the union of two sets: 0 \leq x < a and 1-a < x \leq 1 for some a, 0 < a < 1. No matter how a > 0 is chosen, there is always some finite number of the W_{n} which together with W_{a} covers R_{1}. Of course, R_{1} minus endpoints is not compact and no finite subcollection of the W_{n} in this example will cover it.

THEOREM. Let S be a compact space and let f: S \rightarrow T be a continuous map of S onto a topological space T Then, T is compact.

Proof:

Let \{ O _{a} be a covering of T by open sets. Since f is continuous, each f^{-1}(O_{a}) is an open set in S. There is a finite covering of S by sets of the collection \{ f^{-1}(O_{a})\} and this gives a corresponding finite covering of T by sets of O_{a}. This completes the proof.

COROLLARY. If f is a continuous map of S into T then f(S) is a compact subset of T.

1.7.1 THEOREM

Let S be a compact space and \{ D_{a}\} a collection of closed subsets such that \bigcap_{a}D_{a} is empty. Then there is some finite set D_{a_{1}}, \ldots, D_{a_{n}} such that \bigcap_{i}D_{a_{i}} is empty.

Proof:

The complement of \bigcap_{a}D_{a} is \bigcup_{a}(S-D_{a}); if the intersection set is empty. the union covers S. There is a finite set of indices a_{i} such that S \subset \bigcup_{i}(S - D_{a_{i}}) and conequently \bigcup_{i}D_{a_{i}} is empty for the same finite set of indices.

COROLLARY 1.

Let D_{n}, n \in \mathcal{N} be a sequence of non empty closed subsets of the compact space S with D_{n+1} \subset D_{n}. Then, \bigcap_{n}D_{n} is not empty.

APPLICATION:

The Cantor Middle Third Set D

From R_{1}, “delete” the middle third: \frac{1}{3} < x < \frac{2}{3}. Let D_{1} denote the residue: it is a union of two closed intervals. Let D_{2} denote the closed set in D_{1} complementary to the union of the middle third intervals: \frac{1}{9} < x < \frac{2}{9} and \frac{7}{9} < x < \frac{8}{9}. Continuing inductively, define D_{n} \subset D_{n-1} consisting of 2^{n} closed mutually exclusive intervals Let D = \bigcap_{n}. This is homeomorphic to the space of Example 3 of 1.6.

COROLLARY 2.

A lower semi-continuous (upper semi-continuous) real-valued function on a compact space has finite glb (greatest lower bound), lub (least upper bound) and always attains these bounds at some points of space.

This follows from the preceding corollary and the fact that the set where f(x) \leq r is closed, for every r (similarly, f(x) \geq r.

1.7.2 THEOREM

A topological space with the property: “every collection of closed subsets with empty set-intersection has a finite subcollection whose set-intersection is empty” is compact.

Proof:

The proof, like that of the Theorem of 1.7.1 is based on the duality between open and closed sets.

DEFINITION.

If a point x of a topological space S belongs to an open subset of S whose closure is compact, then S is called locally compact set at x; S i locally compact if it has this property at every point.

COROLLARY.

A closed subset of a locally compact space is locally compact in the induced topology. Similarly, a closed subset of a compact space is compact. The union of a finite number of compact subsets is compact.

Proof: HW.

A set U in a topological space is called a neighbourhood of a point z if there is an open set O such that z \in O \subset U; z is called an inner point of U. A set F is covered by a collection \{ U_{i}\} if each point of F is an inner point of some set U_{i}.

1.7.3.

A space S is called a Hausdorff space if for every x, y \in S, x \neq y, there exist open sets U and V including x and y respectively such that U \bigcap V = \phi where \phi is the empty set; an equivalent property is the existence of a closed neighbourhood of x not meeting y.

HW: Show that a compact subset of a Hausdorff space is closed.

Lemma: Let S be a compact Hausdorff space, let F be a closed set in S, and x a point not in F. Then there is a closed neighbourhood of x, such that W \bigcap F = \Phi

For each y \in F let U_{y} be a neighbourhood of y and W_{y} a neighbourhood of x, such that U_{y} \bigcap W_{y}= \Phi. There is a covering of F by sets U_{y_{1}},\ldots, U_{y_{n}}. Let W_{x} be the intersection of the associated W_{y_{i}} where i = 1, 2, \ldots, n and let W be the closure of W_{x}. The union of the U_{y_{i}} does not meet W. Then W \bigcap F = \Phi which completes the proof.

THEOREM. Let U be a compact Hausdorff space and let F_{n} be a sequence of closed subsets of U. If U is contained in the union of sets F_{n}, then at least one of the sets F_{n} has inner points.

Proof. Take a sequence C_{1} \supset C_{2} \supset \ldots of non empty compact neighbourhoods such that for each n, (\bigcup_{1}^{n}F_{i}) \bigcap C_{n} = \Phi. This leads to \bigcap C_{n}=\Phi a contradiction. QED.

A set is nowhere dense if its closure has no inner points. A space is said to be of the second category if it cannot be expressed as the union of a countable number of nowhere dense subsets. Hence, a compact Hausdorff space is of the second category. Complete metric spaces, to be defined later, are also of the second category.

1.7.4 THEOREM:

Let S be a locally compact space. There exists a compact space S^{*} and a point z in S^{*} such that S^{*}-z is homeomorphic to S.

Proof:

Let z denote a “new” point, not in S, and let S^{*} denote the set union of S and z. If Y is a subset of S, let Y^{*} \subset S^{*} denote the union of Y and z. We topologize S^{*}. Any open set in S is also open in S^{*}. In addition, if X is a compact subset of S and Y is the complement S-X, then Y^{*} is open in S^{*}. These open sets are taken as a sub-basis for open sets in S^{*}.

Suppose now that we have some covering of S^{*} by a family of open sets. Then z belongs to one of these open sets, say z \in U^{*}. The complement of U^{*} is a compact subset of S . Hence, the complement is covered by a finite subset of the given covering sets, because of the compactness in S. Together with U^{*}, this gives a finite covering of S^{*}.

QED.

1.8 Tychonoff Theorem

THEOREM:

Let S_{a} where a \in \{ a \}, be compact spaces and let P be the topological product of the S_{a}. Then, P is compact.

Proof:

Let P = S_{1} \times S_{2} and let F denote a family of open sets of P covering P. For each point 1x_{1} of S_{1}, the closed subset x_{1} \times S_{2} of P is homeomorphic to S_{2} amd is therefore compact. Each point of x_{1} \times S_{2} belongs to a set in F because F is a covering. Because of the way in which a product is topologized, it follows that each point of x_{1} \times S_{2} belongs to some open set U \times V of P such that U \times V is a subset of some set of F. It follows from its compactness that x_{1} \times S_{2} is contained in the union of a finite number of sets

U_{1} \times V_{1}, \ldots, U_{n} \times V_{n}

each of which is a subset of some set of F. Let U^{'} = \bigcap U_{i}. Then, U^{'} \times S_{2} is covered by a finite number of sets of F.

Since x_{1} is an arbitrary point of S_{1} amd S_{1} is compact, there exist a finite number of open sets of S_{1}

U_{1}^{'}, \ldots, U_{m}^{'}

which cover S_{1} and which are such that there is a finite number of sets of F covering U_{1}^{'} \times S_{2} where i=1, \ldots, m. The totality of sets of F thus indicated is a finite number which covers S_{1} \times S_{2}. This completes the proof for the case of two factors. The case for a finite number of factors follows by a simple induction.

EXAMPLE.

Let R_{n} = R_{1} \times R_{1} \times \ldots R_{1}, n factors. Then, R_{n} is compact and it follows that E_{n} = product of n real lines is locally compact.

1.8.1

To consider the general case, let \{ a\} be an arbitrary collection of at least two indices: let S_{a} be compact topological spaces, let P be the topological product, and let F be a collection of open subsets of F covering P. The proof that P is compact is by contradiction. Accordingly, we shall suppose that no finite subcollection of sets of F covers P.

It was shown by Zermelo that it is possible to well-order the set of all subsets of a given set by the use of an axiom-of-choice of appropriate power, namely the cardinal number of the set of all subsets of the given set. A well-ordering of objects permits them to be inspected systematically.

Using such a well-ordering we can enlarge the given family F to a family F^{*} of open sets, where F^{*} has the following properties:

  1. F^{*} is a covering of P by open sets.
  2. No finite subcollection of P by open sets.
  3. If we adjoin to F^{*} any open subset of P not already in F^{*}, then the enlarged collection does contain a finite subcollection which covers P. Of course, it is in (3) that F^{*} has a property not necessarily true of F

Using this enlarged family the proof for the general case becomes similar to the proof for two factors. Let b denote an arbitrary index in \{ a \} which shall be fixed temporarily, and let P_{b} denote the product of all factors S_{a} excepting S_{b}. Then P is homeomorphic to S_{b} \times P_{b}.

Suppose for a moment that to each point x_{b} \times S_{b} there exists an open U_{b} \subset S_{b} containing x_{b} such that

*) U_{b} \times P_{b} \in F^{*}

There must then exist some finite covering of S_{b} by sets U_{b}^{1}, U_{b}^{2}, \ldots, U_{b}^{n} each satisfying *). The producgt P is covered by the union of U_{b}^{i} \times P_{b}, i=1, \ldots, n. This is impossible by the contradiction of F^{*}. Hence in each S_{b} there is at least one x_{b} which does not satisfy the first sentence of this paragraph.

It follows by the axiom of choice that P contains at least one point x = PROD x_{b} such that if U_{b} is an open set in S_{b} and x_{b} \in U_{b} then *) is false. This holds for each coordinate x_{b} of x. This implies for each coordinate x_{b} of x that if x_{b} is in an open set U_{b} of S_{b} then there is a finite collection of sets in F^{*}

O_{b}^{1}, O_{b}^{2}, \ldots, O_{b}^{n_{b}}

which together with U_{b} \times P_{b} forms a covering of P.

The point x belongs to an open set O_{x} \in F^{*}. There is some open set contained in O_{x} which contains x and is of the form

U_{a_{1}} \times U_{a_{2}} \times \ldots U_{a_{n}} \times P_{a_{1}a_{2}a_{3}\ldots a_{n}}

for some finite set of indices a_{i} and where the last set is the product of all S_{n} with the exception of S_{a_{i}}, where i=1,2, \ldots, n. For each a_{i} there exists a finite collection of sets of F^{*} which together with U_{a_{i}} \times P_{a_{i}} covers P, say these sets are

**) O_{a_{i}}^{1}, O_{a_{i}}^{2}, \ldots, O_{a_{i}}^{n_{i}}, where i=1, 2, \ldots, n

Then P is covered by the union of O_{x} and the sets of **). This contradiction completes the proof.

QED.

1.8.2 EXAMPLE

The infinite-dimensional torus described in 1.6 whose “dimension” equals the cardinal power of the set of indices A is compact. It is a commutative group where the addition of two points is carried out by adding the respective coordinates in each factor S_{n} = C_{1} each of these factors being itself a commutative group. The group addition is continuous in the topology and this defines a topological group (1.11) In fact, this is a universal compact commutative topological group (depending on the cardinal power of the group). See for example the following paper: Discrete Abelian groups and their character groups, Ann. of Math., (2) 36 (1935) pp. 71-85.

The principal theorem of this section is due to Tychonoff (see: Uber einen Funktionenraum Math. Ann. III (1935), pp. 762-766). The present proof is dual to a proof given by Bourbaki (see: Topologie generale, Paris, 1942).

1.9 Metric Spaces

DEFINITION.

A set S of points is called a metric space if to each pair x, y \in S there is associated a non-negative real number d(x,y) the distance from x to y, satisfying

  1. d(x,y)=0 if and only if x=y.
  2. d(x,y)=d(y,x)
  3. d(x,y) + d(y,z) \geq d(x,z) where x,y,z \in S.

The distance function d(x,y) also called the metric induces a topology in S as follows. For each r >0 let S_{r}(x) denote the sphere of radius r, that is, the set of y \in S such that d(x,y) <r. Now let S_{r}(x), for all positive r and all x \in S constitute a basis for open sets. This choice of basis makes S a topological space. A space is called metrizable if a metric can be defined for it which induces in it the desired topology. It is clear that a metric space has a countable basis at each point x, namely S_{r}(x) where r is rational.

EXAMPLE 1.

If S_{1} and S_{2} are metric spaces then S_{1} \times S_{2} is a metric space in the metric

d((x_{1}, x_{2}),(y_{1}, y_{2})) = max (d(x_{1},y_{1}), d(x_{2},y_{2}))

where x_{1},y_{1} \in S_{1}, and x_{2}, y_{2} \in S_{2}. The topology determined by this metric is the same as the product topology.

EXAMPLE 2.

The set F of continuous functions defined on a compact space S with values in a metric space M becomes a metric space by defining for f, g \in F

d(f,g) = lub (x \in S) [d_{M}(f(x), g(x))] where d_{M} is the metric in M. See corollary 2, section 1.7.1

THEOREM:

The collection of open sets of a compact metric space S has a countable basis.

Proof:

For each n \in \mathcal{N} there is a covering of the space by a finite number of open sets each of diameter at most \frac{1}{n}. The countable collection of these sets for all n is a basis.

EXAMPLE 3:

If S is a compact metric space and E_{1} denotes the real line, then S \times E_{1} is a metrizable locally compact space with a countable basis for open sets.

By Example 1 above, the space is metrizable. If \{ U_{m}\} and \{ V_{m}\} are countable bases in S and E_{1} respectively then \{ U_{m} \times V_{m}\} forms a countable base in S \times E_{1}. If E_{1n} = \{ x \in E_{1}, |x| \leq n\} then S \times E_{1n} is a compact subset of S \times E_{1} and any point of the product is interior to S \times E_{1n} for n large enough. This proves the local compactness.

1.9.1

The following is of interest:If (x,y) is a metric for a space M then the following equivalent metric:

(x,y)^{'} = \frac{(x,y)}{1+(x,y)} \leq 1

is a bounded metric. Properties (1) and (2) above are obviously satisfied. For (3) , one uses the fact that the function \frac{t}{1+t} increases with t. Thus

(x,y)^{'}+(y,z)^{'} \geq \frac{(x,y)}{1+(x,y)+(y,z)} + \frac{(y,z)}{1+(y,z)+(x,y)} = \frac{(x,y)+(y,z)}{1+(x,y)+(y,z)} \geq \frac{(x,z)}{1+(x,z)} = (x,z)^{'}

This has the following consequence:

Lemma:

Let M be a space which is the union of a system M_{a} where a \in \{ a\} of open mutually exclusive sets. Suppose each M_{a} of open mutually exclusive sets. Suppose each M_{a} is a metric space and carries a metric d_{a} bounded by 1. Define a function d(x,y) which is equal to 2 if x and y are not in the same M_{a}; otherwise let d agree with the appropriate d_{a}. Then d is a metric for M.

Proof: HW.

1.9.2

A sequence of points x_{n} in a metric space is said to converge to a point x, symbolically x_{n} \rightarrow x if \lim d(x,x_{n}) =0. A sequence of points x_{n} satisfies the Cauchy convergence criterion if when \epsilon >0 is given there is an N such that for m,n >N d(x_{n},x_{m})<\epsilon. A metric space is called complete if every sequence of points satisfying the Cauchy criterion converges to a point of the space. A subset of a space is called dense (everywhere dense) in the space if every point of space is a limit of some sequence of points of the subsets.

To be continued in next blog,

Cheers,

Nalin Pithwa

Beginning steps to intuitive physics

Reference: Newtonian Mechanics by A. P. French, The M.I.T. Introductory Physics Series

In the beginning was Mechanics. — Max Von Laue, History of Physics (1950)

I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this — from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena. —- Newton, Preface to the Principia (1686).

  1. What is the order of magnitude of the number of times that the earth has rotated on its axis since the solar system was formed?
  2. During the average lifetime of a human being, how many heart beats are there? How many breaths?
  3. Make reasoned estimates of (a) the total number of ancestors you would have (ignoring inbreeding) since the beginning of the human race, and (b) the number of hairs on your head.
  4. Assume the present world population to be 7 \times 10^{9}.(a) How many square kilometers of land are there per person? How many feet long is the side of a square of that area? (b) If one assumes that the population has been doubling every 50 years, through out the existence of the human race, when did Adam and Eve start it all? If the doubling every 50 years were to continue, how long will it be before people were standing shoulder to shoulder over all the land area of the world?
  5. Estimate the order of magnitude of the mass if (a) a speck of dust (b) a grain of salt (or sugar, or sand) (c) a mouse (d) an elephant (e) the water corresponding to 1 inch of waterfall over 1 square mile, (f) a small hill, 500 ft high; and (g) Mount Everest.
  6. Estimate the order of magnitude of the number of atoms in (a) a pin’s head (b) a human being (c) the earth’s atmosphere and (d) the whole earth
  7. Estimate the fraction of the total mass of earth that is now in the form of living things.
  8. Estimate (a) the total volume of ocean water on the earth and (b) the total mass of salt in all the oceans.
  9. It is estimated that there are about 10^{80} protons in the known universe. If all these were lumped into a sphere so that they were just touching, what would be the radius of the sphere? Ignore the spaces left when spherical objects are packed and take the radius of a proton to be about 10^{-15} m.
  10. The sun is losing mass (in the form of radiant energy) at the rate of about 4 million tons per second. What fraction of its mass has it lost during the lifetime of the solar system?
  11. Estimate the time in minutes that it would take for a theatre audience of about 1000 people to use up 10 % of the available oxygen if the building were scaled. The average adult absorbs about one sixth of the oxygen that he or she inhales at each breath.
  12. Solar energy falls on the earth at the rate of about 2 cal/cm^{2}/minute. Estimate the total amount of power, in megawatts or horsepower, represented by the solar energy falling on an area of 100 square miles — about the area of a good-sized city. How would this compare with the total power requirements of such a city? (1 cal = 4.2Joules; 1W = 1 joule/sec; 1 HP is 746 W).
  13. Starting from an estimate of the total mileage that an automobile tire will give before wearing out, estimate what thickness of rubber is worn off during one revolution of the wheel. Consider the possible physical significance of the result.
  14. An inexpensive wristwatch is found to lose 2 minutes per day. (a) what is the fractional deviation from the correct rate? (b) By how much could the length of the ruler (nominally 1 feet long) differ from exactly 12 inches and still be fractionally as accurate as the watch?
  15. The astronomer Tycho Brahe made observations on the angular positions of stars and planets by using a quadrant, with one peephole at its centre of curvature and peephole mounted on the arc. One such quadrant had a radius of about 2 inches, and Tycho’s measurements could usually be treated to 1 minute of arc. What diameter of peepholes would have been needed for him to attain this accuracy?
  16. Jupiter has a mass about 300 times that of the earth, but its mean density is only about one-fifth that of the earth. (a) What radius would a planet of Jupiter’s mass and earth’s density would have? (b) What radius would a planet of earth’s mass and Jupiter’s density have?
  17. Identical spheres of material are tightly packed in a given volume of space. (a) Consider why one does not need to know the radius of the spheres, but only the density of the material, in order to calculate the total mass contained in the volume, provided that the linear dimensions of the volume are large compared to the radius of the individual spheres. (b) Consider the possibility of packing more material if two sizes of spheres may be chosen and used. (c) Show that the total surface of the spheres of part (a) does depend on the radius of the spheres (an important consideration in the design of such things as filters, which absorb in proportion to the total exposed surface area within a given volume)
  18. Calculate the ratio of surface area to volume for (a) a sphere of radius r (b) a cube of edge a, and (c) a right circular cylinder of diameter and height both equal to d. For a given value of the volume, which of these shapes has the greatest surface area? The least surface area ?
  19. How many seconds of arc does the diameter of the earth subtend at the sun? At what distance from an observer should a football be placed to subtend an equal angle?
  20. From the time the lower limb of the sun touches the horizon it takes approximately 2 min for the sun to disappear beneath the horizon. (a) Approximately what angle (expressed both in degrees and in radians) does the diameter of the sun subtend at the earth? (b) At what distance from your eye does a coin of about 0.75 inches diameter (e.g. a dime or a nickel) just block out the disk of the sun?
  21. What solid angle in steridians does the sun subtend at the earth?
  22. How many inches per mile does a terrestrial great circle (e.g. a meridian of longitude) deviate from a straight line?
  23. A crude measure of the roughness of a nearly spherical surface could be defined by \triangle {r}/r, where \triangle {r} is the height or depth of local irregularities. Estimate this ratio for an orange, a ping-pong ball, and the earth.
  24. What is the probability (expressed as 1 chance in 10^{n}) that a good-sized meteorite falling to the earth would strike a man-made structure? A human?
  25. Two students want to measure the speed of sound by the following procedure. One of them, positioned some distance away from the other, sets off a firecracker. The second student starts a stopwatch when he sees the flash and stops it when he hears the bang. The speed of sound in air is roughly 300m per second, and the students must admit the possibility of an error (of undetermined sign) of perhaps 0.3 seconds inthe elapsed time recorded. If they wish to keep the error in the measured speed of sound to within 5% what is the minimum distance over which they can perform the experiment?
  26. A right triangle has sides of length 5m, 1 m adjoining the right angle. Calculate the length of the hypotenuse from the binomial expansion to two terms only, and estimate the fractional error in this approximate result.
  27. The radius of a sphere is measured with an uncertainty of 1 percent. What is the percentage uncertainty in the volume?
  28. Construct a piece of semilogarithmic graph paper by using the graduations on your slide rule to mark off the ordinates and a normal ruler to mark off the abscissa. On this piece of paper draw a graph of the function y=2^{x}.
  29. The subjective sensations of loudness or brightness have been judged to be approximately proportional to the logarithm of the intensity, so that equal multiples of intensity are associated with equal arithmetic increases in sensation (For example, intensities proportional to 2, 4, 8, and 16 would correspond to equal increases in sensation). In acoustics, this has led to the measurement of sound intensities in decibels. Taking as a reference value the intensity I_(0) of the faintest audible sound, the decibel level of a sound of intensity I is defined by the equation: dB = 10 \times \log_{10}{\frac{I}{I_{0}}}. (a) An intolerable noise level is represented by about 120 dB. By what factor does the intensity of such a sound exceed the threshold intensity I_{0}? (b) A simple logarithmic scale is used to describe the relative brightness of stars (as seen from the earth) in terms of magnitudes. Stars differing by “one magnitude” have a ratio of apparent brightness equal to about 2.5 Thus, a “first magnitude” (very bright) star is 2.5 times brighter than a second magnitude star (2.5)^{2} times brighter than a third-magnitude star, and so on. (These differences are due to largely differences of distance). The faintest stars detectable with the 200 inches Palomar telescope are of about the twenty-fourth magnitude. By what factor is the amount of light reaching us from such a star less than we receive from a first magnitude star?
  30. The universe appears to be undergoing a general expansion in which the galaxies are receding from us at speeds proportional to their distances. This is described by Hubble’s Law, v= \alpha \times r where the constant \alpha corresponds to v becoming equal to speed of light, c = 3 \times 10^{8} meters per second at r \approx 10^{26} metres. This would imply that the mean mass per unit volume in the universe is decreasing with time. (a) Suppose that the universe is represented by a sphere of volume V at any instant. Show that the fractional incfease of volume per unit time is given by \frac{1}{V} \times \frac{dV}{dt} =3\alpha (b) Calculate the fractional decrease of mean density per second and per century.
  31. The table lists the mean orbit radii of successive planets expressed in terms of the earth’s orbit radius. The planets are numbered in order (n) \begin{array}{ccc}n & planet & r/r_{s} \\ 1 & mercury & 0.39 \\ 1 & venus & 0.72 \\3 & earth & 1.00 \\ 4 & Mars & 1.52 \\ 5 & Jupiter & 5.20 \\ 6 & Saturn & 9.54 \\7 & Uranus & 19.2  \end{array}

(a) Make a graph in which \log{\frac{r}{r_{g}}} is ordinate and the number n is abscissa. (Or, alternatively, plot values of \frac{r}{r_{g}} against n on semilogarithmic paper). On this same graph, replot the points for Jupiter, Saturn, and Uranus at values of n increased by unity (that is, at n=6, 7, and 8). The points representing the seven planets can then be reasonably well fitted by a straight line.

(b) If n=5 in the revised plot is taken to represent the asteroid belt between the orbits of Mars and Jupiter, what value of \frac{r}{r_{g}} would your graph imply for this ? Compare with the actual mean radius of the asteroid belt.

(c) If n=9 is taken to suggest an orbit radius for the next planet (Neptune) beyond Uranus, what value of \frac{r}{r_{g}} would your graph imply? Compare with the observed value.

(d) Consider whether in the light of (b) and (c) your graph can be regarded as the expression of a physical law with predictive value. (As a matter of history, it was so used.)

Approximations:

Binomial Theorem:

for x << 1, (1+x)^{n} \approx 1+nx

For example, (1+x)^{3} \approx 1+3x

For example, (1-x)^{\frac{1}{2}} \approx 1- \frac{1}{2}x \approx (1+x)^{-\frac{1}{2}}

For b << a, (a+b)^{n} \approx a^{n}(1+\frac{b}{a}) \approx a^{n}(1+n\frac{b}{a})

Other expansions:

For \theta << 1 radians, \sin{theta} \approx \theta - \frac{\theta^{3}}{6} tends to \theta

for \theta <<1 radians, \cos{\theta} \approx 1 - \frac{\theta^{2}}{2} tends to 1

For x << 1 \log_{e}{1+x} \approx x

For x <<1, \log_{10}{1+x} \approx 0.43x

Cheers, cheers, cheers,

Nalin Pithwa

Is Math really abstract? I N Herstein answers…

Reference: Chapter 1: Abstract Algebra Third Edition, I. N. Herstein, Prentice Hall International Edition:

For many readers/students of pure mathematics, such a book will be their first contact with abstract mathematics. The subject to be discussed is usually called “abstract algebra,” but the difficulties that the reader may encounter are not so much due to the “algebra” part as they are to the “abstract” part.

On seeing some area of abstract mathematics for the first time,be it in analysis, topology, or what not, there seems to be a common reaction for the novice. This can best be described by a feeling of being adrift, of not having something solid to hang on to. This is not too surprising, for while many of the ideas are fundamentally quite simple, they are subtle and seem to elude one’s grasp the first time around. One way to mitigate this feeling of limbo, or asking oneself “What is the point of all this?” is to take the concept at hand and see what it says in particular concrete cases. In other words, the best road to good understanding of the notions introduced is to look at examples. This is true in all of mathematics.

Can one, with a few strokes, quickly describe the essence, purpose, and background for abstract algebra, for example?

We start with some collection of objects S and endow this collection with an algebraic structure by assuming that we can combine, in one or several ways (usually two) elements of this set S to obtain, once more, elements of this set S. These ways of combining elements of S we call operations on S. Then we try to condition or regulate the nature of S by imposing rules on how these operations behave on S. These rules are usually called axioms defining the particular structure on S. These axioms are for us to define, but the choice made comes, historically in mathematics from noticing that there are many concrete mathematical systems that satisfy these rules or axioms. In algebra, we study algebraic objects or structures called groups, rings, fields.

Of course, one could try many sets of axioms to define new structures. What would we require of such a structure? Certainly we would want that the axioms be consistent, that is, that we should not be led to some nonsensical contradiction computing within the framework of the allowable things the axioms permit us to do. But that is not enough. We can easily set up such algebraic structures by imposing a set of rules on a set S that lead to a pathological or weird system. Furthermore, there may be very few examples of something obeying the rules we have laid down.

Time has shown that certain structures defined by “axioms” play an important role in mathematics (and other areas as well) and that certain others are of no interest. The ones we mentioned earlier, namely, groups, rings, fields, and vector spaces have stood the test of time.

A word about the use of “axioms.” In everyday language, “an axiom means a self-evident truth”. But we are not using every day language; we are dealing with mathematics. An axiom is not a universal truth — but one of several rules spelling out a given mathematical structure. The axiom is true in the system we are studying because we forced it to be true by “force” or “our choice” or “hypothesis”. It is a licence, in that particular structure to do certain things.

We return to something we said earlier about the reaction that many students have on their first encounter with this kind of algebra, namely, a lack of feeling that the material is something they can get their teeth into. Do not be discouraged if the initial exposure leaves you in a bit of a fog.Stick with it, try to understand what a given concept says and most importantly, look at particular, concrete examples of the concept under discussion.

Follow the same approach in linear algebra, analysis and topology.

Cheers, cheers, cheers,

Nalin Pithwa

Some simply astounding facts shown by basic topology

Consider the Jordan curve theorem. A simple closed curve (that is, the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary.

The beginner/reader/student might say but so what …this is too obvious…is topology used to prove such silly obvious results only?

Well, be that as it may, even these “obvious” things are to be taken much more seriously: one can easily give examples of propositions which sound as “obvious” as the Jordan curve theorem, but which may be proved false. Who would believe, for example, that in a plane there are three (four, five, …in fact, infinitely many) simply connected bounded regions which all have the same boundary or that one can find in three dimensional space a simple Jordan arc (that is, a topological image of a polygonal line) such that there are circles outside of this arc that cannot possibly be contracted to a point without meeting it? There are also closed surfaces of genus zero which possess an analogous property. In other words, one can construct a topological image of a sphere and an ordinary circle in its interior in such a way that the circle may not be contracted to a point wholly inside the surface?

Cheers, cheers, cheers,

Nalin Pithwa

The tiling theorem for n=2

For n=2, the tiling theorem states that if a country is divided into sufficiently small provinces, then there necessarily exist points at which at least three provinces come together. Here these provinces may have entirely arbitrary shapes; in particular, they need not even be connected; each one may consist of several pieces.

Topology bare facts: part 6

NB: same reference: Elementary Concepts of Topology: Paul Alexandroff

A:

The proof of the theorem of the invariance of Betti numbers which was presented last, following Alexander and Hopf, is an application of the general method of approximation of continuous mappings of polyhedra by simplicial mappings. We wish to say here a few more words about this method. Let f be a continuous mapping of a polyhedra P^{'} into a polyhedron P^{''}, and let the complexes K^{'} and K^{''} be simplicial decomposition of the polyhedra P^{'} and P^{''} respectively. Let us consider a subdivision K_{1}^{''} of K^{''} so fine that the simplexes and the barycentric stars of K_{1}^{''} are smaller than a presribed number \epsilon; then, we choose the number \delta so small that the two arbitrary points of P^{'} which are less than \delta apart go over by means of f into points of P^{''} whose separation is less than the Lebesgue number \sigma of the barycentric covering of K_{1}^{''}. Now consider a subdivision K_{1}^{'} of K^{'} whose simplexes are smaller than \delta. The images of the vertex frames of K_{1}^{''} have a diameter less than \sigma, and their totality can be considered as an abstract complex Q; because of the smallness of the simplexes of Q, one can apply to this complex the following procedure: one can map it by means of a canonical displacement g in to the complex K_{1}^{"}. The transition of K_{1}^{'} to Q and the map g from Q to g(Q) together produce a simplicial mapping f_{1} of K_{1}^{'} into K_{1}^{''}. This mapping (considered as a mapping from P^{'} into P^{''} ) differs from f by less than \epsilon (that is, for every point a of P^{'} the distance between the points f(a) and f_{1}(a) is less than \epsilon). The mapping f_{1} is called a simplicial approximation of the continuous mapping f (and, indeed, one of fineness \epsilon).

By means of the mapping f_{1} there corresponds to each cycle z of K^{'} (where z is to be regarded as belonging to the subdivision K_{1}^{'} of K^{'}) a cycle f_{1}(z) pf K_{1}^{''}. Moreover, one can easily convince oneself that if z_{1} \sim z_{2} in K^{'} then it follows that f_{1}(x_{1}) \sim f_{2}(x_{2}) in K_{1}^{''} so that to a class of homologous cycles of K^{'} there corresponds a class of homologous cycles of K_{1}^{''}. In other words, there is a mapping of the Betti groups of K^{'} into the corresponding Betti groups of K_{1}^{''}; since the mapping preserves the group operation (additIon), it is, in the language of algebra, a homomorphism. But there also exists a uniquely determined isomorphism between the Betti groups of K_{1}^{''} and K_{''}, so that as a result, we obtain a homomorphic mapping of the Betti groups of K^{'} into the corresponding groups of K^{''}.

Consequently, we have the following fundamental theorem (first formulated by Hopf):

A continuous mapping f of a polyhedron P^{'} into a polyhedron P^{''} induces a uniquely determined homeomorphic mapping of all the Betti groups of the simplicial decomposition K^{'} of P^{'} into the corresponding groups of the simplicial decomposition K^{''} of P^{''}.

If the continuous mapping f is one-to-one (therefore, topological) it induces an isomorphic mapping of the Betti groups of P^{'} onto the corresponding Betti groups of P^{''}.

By this theorem a good part of the topological theory of continuous mappings of polyhedra (in particular of manifolds) is reduced to the investigation of the homomorphisms induced by these mappings, and thus to considerations of a purely algebraic notion. In particular, one arrives at far reaching results concerning the fixed points of a continuous mapping of a polyhedron onto itself.

(Note: We mean here principally the Lefschetz-Hopf fixed point formula which completely determines (and indeed expresses by algebraic invariants of the above homomorphism) the so-called algebraic number of fixed points of the given continuous mapping (in which every fixed point is to be counted with a definite multiplicity which can be positive, negative or zero.))

B.

We close our topic of topological invariance theorems with a few remarks about the general concept of dimension which are closely related to the ideas involved in the previous invariance proofs. Our previous considerations have paved the way for the following definition:

A continuous mapping f of a closed set F of \mathcal{R}^{n} onto a set of F^{'} lying on the same \mathcal{R}^{n} is called an \epsilon-transformation of the set F (into the set F^{'}) if every point a of F is at a distance less than \epsilon from its image point f(a).

We now present the proof of the following theorem, which to a large extent justifies the general concept of dimension from the intuitive geometrical standpoint, and allows the connection between set-theoretic concepts and the methods of polyhedral topology to be more easily and simply understood than do the brief and, for many tastes, too abstract remarks concerning projection spectra:

Transformation theorem:

For each \epsilon>0, every r-dimensional set F can be mapped continuously onto an r-dimensional polyhedron by means of an \epsilon-transformation; on the other hand, for sufficiently small \epsilon, there is no \epsilon-transforation of F into a polyhedron whose dimension is at most r-1.

The proof is based on the following remark. If

I: F_{1}, F_{2}, \ldots, F_{s}

is an \epsilon-covering of F, then the nerve of the system of sets I is defined first as an abstract complex: to each set F_{i}, where 1 \leq i \leq s let there correspond a “vertex” a_{i} and consider a system of vertices

a_{i_{0}}, a_{i_{1}}, \ldots, a_{i_{s}}

as the vertex frame of a simplex (of the nerve Kof I) if and only if the sets F_{i_{0}}, F_{i_{1}}, \ldots, F_{i_{r}} have a non empty intersection. However, one can realize this abstract complex geometrically if one chooses for a_{i} a point of F_{i} itself, or a point from an arbitrarily prescribed neighbourhood of F_{i}, and then allows the vertex frame of the nerve to be spanned by the ordinary geometrical simplexes.. This construction is always possible, and yields as the nerve of the system of sets I an ordinary geometrical polyhedral complex provided the coordinate space \mathcal{R}^{n} in which F lies is of high enough dimension, (note:) but this condition can always be satisfied because one can, if need be, imbed the \mathcal{R}^{n} in which F lies in a coordinate space of higher dimension.

C:

In any case, we now assume that a_{i} is at a distance less than \epsilon for each F_{i} and prove the following two lemmas:

C1:

If K is a geometrically realized nerve of the \epsilon-covering I of F, then every complex Q whose vertices belong to F, and whose simplexes are smaller than the Lebesgue number \sigma of the covering I, goes over into a subcomplex of K by means of a 2\epsilon-displacement of its vertex.

Indeed, associate to each vertex b of Q as the point f(b) one of those vertices a_{i} of K which correspond to the sets F_{i} containing the point b. Thereby, a simplicial mapping f_{i} of Q into K is determined; since, the distance between a and f(a) is clearly less than 2\epsilon our lemma is proved. QED.

C2:

The conclusion of lemma C! also holds (with 3\epsilon in place of 2\epsilon) if hte vertices of Q do not necessarily belong to F but if one knows that they lie at a distance of less than 1/3 \sigma from F, and that the diameters of the simplexes Q do not exceed the number 1/3 \sigma.

In order to reduce this lemma to the preceding one, it is only necessary to transform the vertices of Q into points of P by means of a 1/3 \sigma displacement.

We now decompose the \mathcal{R}^{n} into simplexes which are smaller than 1/3 \sigma, and denote by Q the complex which consists of all those simplexes which contain points of F in their interiors or on their boundaries; then apply to this complex the lemma just proved. This gives us the following:

A sufficiently small polyhedral neighbourhood Q of F is transformed by means of a 2\epsilon transformation into a polyhedron P, consisting of simplexes of K.

Since F was r-dimensional and the dimension of the nerve of a system of sets is always 1 less than the order of the system of sets, we may assume that P is at most r-dimensional. From the fact that a certain neighbourhood of F is transformed onto the polyhedron P by the 2\epsilon-transformation in question, it follows that F itself will be mapped onto a proper or improper subset of P (that is, in P_{r}).

Thus, we have proved: For every \epsilon>0 F can be mapped onto a subset \Phi of an r-dimensional polyhedron by an \epsilon-transformation.

We now consider a simplicial decomposition K of P whose elements are smaller than \epsilon. Since \Phi is closed, there exists — if \Phi \neq P — an r-dimensional simplex x^{r} of K which contains a homothetic simplex x_{0}^{r} free of points of \Phi. If one now allows the domain x^{r} - x_{0}^{r} which lies between the boundaries of x^{r} and x_{0}^{r} to contract to the boundary of x^{r}, then all the points of \Phi contained in x^{r}, and the points of the set \Phi will be “swept out” of the interior of the simplex x^{r}. By a finite number of repetitions which do not belong to \Phi will be freed of points of the set. One continues the process with (r-1)-dimensional simplexes, and so on. The procedure ends with a polyhedron composed of simplexes, and so on. The procedure ends with a polyhedron composed of simplexes (of different dimensions) of K. \Phi is mapped onto this polyhedron by means of a continuous deformation in which no point of \Phi leaves that simplex of K to which it originally belonged; consequently, every point of \Phi is displaced by less than \epsilon. Hence, the whole passage from F to P is a 2\epsilon-transformation of the set F so that the first half of our theorem is proved.

QED.

In order to prove the second half, prove the following more general statement: there exists a fixed number \epsilon(F)>0 such that the r-dimensional set F can be mapped by an \epsilon(F)-transformation into no set whose dimension is at most (r-1).

We assume that there is no such \epsilon(F). Then, for every \epsilon>0 there exists a set F_{\epsilon} of dimension at most (r-1) into which F can be mapped by means of an <\epsilon-transformation. Consider an \epsilon-covering of the set F_{\epsilon}

II \ldots F_{1}^{\epsilon}, F_{2}^{\epsilon}, \ldots, F_{s}^{\epsilon} of order less than or equal to r, and denote by F_{s} the set of all points of F which are mapped into F_{i}^{\epsilon} by our transformation. Clearly, the sets F_{i} form a 3\epsilon-covering of F of the same order as (II), therefore of order less than or equal to r. Since this holds for all \epsilon we must have dim F less than or equal to (r-1), which contradicts our assumption. With this, the transformation theorem is completely proved.

QED.

D:

Remark:

If the closed set F of R^{n} has no interior points, then for every \epsilon it may be \epsilon transformed into a polyhedron of dimension at most (n-1); it suffices to decompose the \mathcal{R}^{n} into \epsilon-simplexes and to “sweep out” each n-dimensional simplex of this decomposition. A set without interior points is thus at most (n-1)-dimensional. Since, on the other hand, a closed set of \mathcal{R}^{n} which possesses interior points is necessarily n-dimensional (indeed, it contains n-dimensional simplexes!), we have proved the following:

A closed subset of \mathcal{R}^{n} is n-dimensional if and only if it contains interior points.

With this we close our sketchy remarks on the topological invariance theorems and the general concept of dimension — the reader will find a detailed presentation of the theories dealing with these concepts in literature and above all in the book of Paul Alexandroff and Herr Hopf.

E:

Examples of Betti groups:

E1:

The one-dimensional Betti group of the circle as well as of the plane annulus is the infinite cyclic group; that of the lemniscate is the group of all linear forms u\zeta_{1} + v \zeta_{2} (with integral u and v).

E2:

The one dimensional Betti number of a (p+1)-fold connected plane region equals p.

E3:

A closed orientable surface of genus p has for its one-dimensional Betti group the group of all linear forms:

\sum_{i=1}^{p}u_{i}\xi_{i} + \sum_{i=1}^{p}v_{i}\eta_{i}….with integral u^{i}, v^{i}

Here one takes as generators \xi_{i} and \eta_{i} the homology classes of the 2p canonical closed curves. (For example, in the text Geometry and Imagination by David Hilbert and Cohn-Vossen)

The non-orientable closed surfaces are distinguished by the presence of a non-vanishing one-dimensional torsion group, where by torsion group (of any dimension) we mean the subgroup of the full Betti group consisting of all elements of finite order. The one-dimensional Betti number of a non-orientable surface of genus p is (p-1).

The two dimensional Betti numbers of a closed surface equals 1 or 0 according as the surface is orientable or not. The analogous assertion also holds for the n-dimensional Betti number of an n-dimensional closed manifold.

E4:

Let P be spherical shell and Q be the region enclosed between two coaxial surfaces. The one dimensional Betti number of P is 0, the one dimensioinal Betti number of Q is 2, while the two-dimensional Betti numbers of P and Q have the value 1.

E5:

One can choose as generators of the one-dimensional Betti group of the three dimensional torus the homology classes of the three cycles z_{1}^{1}, z_{2}^{1}, z_{3}^{1} which are obtained from the three axes of the cube by identifying the opposite sides. As generators of the two dimensional Betti group, we can use the homology classes of the three tori into which the three squares through the centre and parallel to the sides are transformed under identification. Therefore, the two Betti groups are isomorphic to one another; each has three independent generators, hence, three is both the one- and two-dimensional Betti number of the manifold.

E6:

For the one- as well as the two-dimensional Betti group of the manifold S^{2} \times S^{3} we have the infinite cyclic group (the corresponding Betti numbers are therefore equal to 1). As z_{0}^{1} choose the cycle which arises from the line segment aa^{'} under the identification of the two spherical surfaces, and as z_{0}^{2}, any sphere which is concentric with the two spheres S^{2} and s^{2} and lies between them.

It is no accident that in the last two examples the one- and two-dimensional Betti numbers of the three-dimensional manifolds in question are equal to one another; indeed, we have the more general theorem, known as the Poincare duality theorem, which says that in an n-dimensional closed orientable manifold, the r- and the (n-r)-dimensional Betti numbers are equal, for 0 \leq r \leq n. The basic idea of the proof can be discerned in the above examples: it is the fact that one can choose for every cycle z^{r} which is not \approx 0 in M^{n} a cycle z^{n-r} such that the so-called “intersection number” of these cycles is different from zero.

E7:

The product of the projective plane with the circle is a non-orientable three-dimensional manifold M^{3}. It can be represented as a solid torus in which one identifies, on each meridian circle, diametrically opposite pairs of points. The one-dimensional Betti number of M^{3} is 1 (every one dimensional cycle is homologous to a multiple of the circle which goes around through the centre of the solid torus); the two-dimensional Betti group (the torus with the aforementioned identification indeed does not bound, but is a boundary divisor of order 2) (Note: the r-dimensional torsion group T_{r}(K) of a complex K is the finite group which consists of all elements of finite order of the Betti group H_{r}(K). The factor group H_{r}(K)/T_{r}(K) is isomorphic to F_{r}(K)). Here again there is a general law; the (n-1)-dimensional torsion group of a closed non-orientable n-dimensional manifold is always a finite group of order 2, while an orientable M^{n} has no (n-1)-dimensional torsion. One can also see from our example that for non-orientable closed manifolds Poincare’s duality theorem does not hold in general.

F:

If we consider the polyhedra mentioned in examples 1, 2, and 3 as polyhedra of three-dimensional space, we notice immediately that both the polyhedron and the region complementary to it in \mathcal{R}^{3} have the same one-dimensional Betti numbers. This can be seen most easily if one chooses as generators of the group H_{1}(P) the homology classes of the cycles x_{1}^{1} and x_{2}^{1} respectively, x^{1} and y^{1}, and as the generators of the group H_{1}(R^{3}-P) the homology classes of the cycles Z_{1}^{1} and Z_{2}^{1} respectively, X^{1} and Y^{1}. This remarkable fact is a special case of one of the most important theorems of all topology, the Alexander duality theorem: the r-dimensional Betti number of an arbitrary polyhedron lying in \mathcal{R}^{n} is equal to the (n-r-1)-dimensional Betti number of its complementary region R^{n}-P for 0 < r < n-1.

The proof of Alexander’s duality theorem is based on the fact that for every x^{r} not \approx0 in P, there exists a x^{n-r-1} in R^{n}-P which is linked with it —- an assertion whose intuitive sense is made sufficiently clear by drawing diagrams. This fact also hols for r=n-1 (since pairs of points which are separated by the (n-1)-cycle concerned appear as zero-dimensional linked cycles. From those considerations the theorem easily follows that the number of regions into which a polyhedron decomposes R^{n} is1 larger thanthe (n-1)- dimensional Betti number of the polyhedron — a theorem which contains the n-dimensional Jordan theorem as a special case. Both this decomposition theorem and the Alexander duality theoremhold for curved polyhedra.

G:

I have intentionally placed in the centre of the presentation those topological theorems and questions which are based upon the concepts of teh algebraic complex and its boundary first, because today this branch pf topology — as no other — lies before us in such clarity that is is worthy of the attention of the widest mathematical circles; second, because since the work of Poincare it is assuming an incresingly more prominent position within topology. Indeed, it has turned out that a larger and larger part of topology is generated by the concept of homology. This holds true especially for the theory of continuous mappings of manifolds, which in recent years — principally through the work of Lefschetz and Hopf — has shown a significant advance to a large extent, this advance has been made possible by the reduction of a series of important questions to the algebraic investigations of the homomorphisms of the Betti group induced by continuous mappings. Recently, the development of set theoretic topology, especially that of dimension theory, has taken a similar turn; it is now known that the concepts of cycle, boundary, Betti groups, etc. hold not onlyfor polyhedra, but also can be generalized to include the case of arbitrary closed sets. Naturally, the circumstances here are much more complicated, but even in these general investigations we have now advanced so far that we are at the beginning of a systematic and entirely geometrical oriented theory of the most general structures of space, a theory which has its own significant problems and its own difficulties. This theory is also based principally on the concept of homology.

Finally, the part of topology which is concerned with the concept of cycle and homology as the part on which the applications of topology depend almost exclusively; the first applications to differential equations, mechanics, and algebraic geometry lead back to Poincare himself. In the last few years, have been increasing almost daily. It suffices here to mention, for example, the reduction of numerous analytical existence proofs to topological fixed point theorems, the founding of enumerative geometry by Van der Waerden, the pioneering work of Lefschetz in the field of algebraic geometry, the investigation of Birkhoff, Morse and others in the calculus of variations in the large, and numerous differential geometrical investigations of others, etc. One may say, without exaggeration, ANYONE WHO WISHES TO LEARN TOPOLOGY WITH AN INTEREST IN ITS APPLICATIONS MUST START WITH BETTI GROUPS, because today, just as in the time of Poincare, most of the threads which lead from topology to the rest of mathematics and bind topological theorems together into a recognizable whole lead through this point.

Cheers, cheers, cheers, 🙂

Nalin Pithwa