Some basic facts about connectedness and compactness

Reference: Hocking and Young’s Topology, Dover Publishers. Chapter 1: Topological Spaces and Functions.

Definition : Separated Space: A topological space is separated if it is the union of two disjoint, non empty, open sets.

Definition: Connected Space: A topological space is connected if it is not separated.

PS: Both separatedness and connectedness are invariant under homeomorphisms.

Lemma 1: A space is separated if and only if it is the union of two disjoint, non empty closed sets.

Lemma 2: A space S is connected if and only if the only sets in S which are both open and closed are S and the empty set.

Theorem 1: The real line E^{1} is connected.

Theorem 2: A subset X of a space S is connected if and only if there do not exist two non empty subsets A and B of X such that X = A \bigcup B, and such that (\overline{A} \bigcap B) \bigcup (A \bigcap \overline{B}) is empty.

Note the above is Prof. Rudin’s definition of connectedness.

Theorem 3: Suppose that C is a connected subset of a space S and that \{ C_{\alpha}\} os a collection of connected subsets of S, each of which intersects C. Then, S = C \bigcup (\bigcup_{\alpha}C_{\alpha}) is connected.

Corollary of above: For each n, E^{n} is connected.

Theorem 4:

Every continuous image of a connected space is connected.

Lemma 3: For n>1, the complement of the origin in E^{n} is connected.

Theorem 5: For each n>0, S^{n} is connected.

Theorem 6: If both f: S \rightarrow T and g: T \rightarrow X are continuous, then the composition gf is also continuous.

Lemma 4: A subset X of a space S is compact if and only if every covering of X by open sets in S contains a finite covering of X.

Theorem 7: A closed interval [a,b] in E^{1} is compact.

Theorem 8: Compactness is equivalent to the finite intersection property.

Theorem 9: A compact space is countably compact.

Theorem 10: Compactness and countable compactness are both invariant under continuous transformations.

Theorem 11: A closed subset of a compact space is compact.

Cheers,

Nalin Pithwa.

Some basic facts about continuity

Reference: (1) Topology by Hocking and Young especially chapter 1 (2) Analysis — Walter Rudin.

Definition 1: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

Definition 2: We may also state the continuity requirement on f as follows: if p is a limit point of \overline{X}, then f(p) is a point of \overline{f(X)}.

Theorem 1: If S is a set with the discrete topology and f: S \rightarrow T any transformation of S into a topologized set T, then f is continuous.

Theorem 2: A real-valued function y=f(x) defined on an interval [a,b] is continuous provided that if a \leq x_{0} \leq b and \epsilon >0, then there is a number \delta >0 such that if |x-x_{0}|<\delta, x in [a,b], then |f(x) - f(x_{0})|< \epsilon. (NB: this is same as definition 1 above).

Theorem 3: Let f: S \rightarrow T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image f^{-1}(O) is open in S.

Theorem 4:

A necessary and sufficient condition that the transformation f: S \rightarrow T of the space S into the space T be continuous is that if x is a point of S, and V is an open subset of T containing f(x), then there is an open set, U in S containing x and such that f(U) lies in V.

Theorem 5:

A one-to-one transformation f: S \rightarrow T of a space S onto a space T is a homeomorphism, if and only if both f and f^{-1} are continuous.

Theorem 6:

Let f: M \rightarrow N be a transformation of the metric space M with metric d into the metric space N with metric \rho. A necessary and sufficient condition that f be continuous is that if \epsilon is any positive number and x is a point of M, then there is a number \delta >0 such that if d(x,y)< \delta, then \rho(f(x), f(y)) < \epsilon.

Theorem 7:

A necessary and sufficient condition that the one-to-one mapping (that is, a continuous transformation) f: S \rightarrow T of the space S onto the space T be a homeomorphism is that f is interior. (NB: A transformation f: S \rightarrow T of the space S into the space T is said to be interior if f is continuous and if the image of every open set subset of S is open in T).

Regards,

Nalin Pithwa.

Ex: 1-7, relation between homeomorphism and continuity

Reference: Exercise: 1-7. Chapter 1. Topology by Hocking and Young.

Prove that a one-to-one transformation f: S \rightarrow T of a space S onto a space T is a homeomorphism if and only if both f and f^{-1} are continuous.

Proof:

Consider the following defintion:

Definition 1-1: A real point p is a limit point of a set X of real numbers if for any positive number \epsilon there exists a point x \in X such that 0 < |p-x| < \epsilon.

Definition of homeomorphism: A homeomorphism of S onto T is a one-to-one transformation f: S \rightarrow T which is onto and such that a point p is a limit point of a subset X of S if and only if f(p) is a limit point of f(X).

But, the above two definitions when combined mean the following: at least for the case of a Euclidean space: A real valued function y=f(x) defined on an interval [a,b] is continuous provided that if a \leq x_{0} \leq b and \epsilon >0, then there is a number \delta >0 such that if |x-x_{0}|<\delta, x in [a,b], then |f(x) - f(x_{0})|<\epsilon.

The above sub-case settles the proof for the Euclidean space E^{1}.

Now, for the more general transformation f: S \rightarrow T, consider definition 1-1 above and the following two theorems (both being equivalent definitions of continuous functions):

Theorem 1-6: Let f: S \rightarrow T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image f^{-1}{O} is open in S.

Next, theorem 1-7: A necesssary and sufficient condition that the transformation f: S \rightarrow T of the space S into the space T be continuous is that if x is a point of S, and V is an open subset of T containing f(x), then there is an open set, U in S containing x such that f(U) lies in V.

Clearly, the above settles the claim for a general transformation which is one-to-onto that it is a homeomorphism if and only if both f and f^{-1} are continuous.

QED.

Your comments are welcome !

Regards,

Nalin Pithwa.

Chapter 1: Topology, Hocking and Young: A theorem due G.D. Birkhoff

Prove that the collection of all topologies on a given set S constitutes a lattice under the partial ordering of finer/coarser topology of a set.

Proof: NB. This theorem has already been proven by eminent mathematician G. D. Birkhoff.

PS: I am making my own attempt here. Your comments are most welcome !

Part 1: TST: Coarser/finer is a partial ordering of topologies.

To prove axiom 1: this order is reflexive.

Let \{ O_{\alpha}\} and \{ R_{\beta} \} be two collections of subsets of a set S, both satisfying the three defining axioms O_{1}, O_{2} and O_{3} of a topology. Let S have two topologies. We will say that the topology \mathcal{T}_{1} determined by \{ O_{\alpha} \} is a finer topology than the topology \mathcal{T}_{2} determined by \{ R_{\beta} \} if every set R_{\beta} is a union of sets O_{\alpha}, that is, each R_{\beta} is open in the \mathcal{T}_{1} topology. We will denote this situation with the symbol \mathcal{T}_{1} \geq \mathcal{T}_{2}.

Axiom 1: reflexive clearly holds because \mathcal{T}_{1} \geq \mathcal{T}_{1} as set inclusion/containment is reflexive.

Axiom 2: antisymmetric also holds true because clearly \mathcal{T}_{1} \geq \mathcal{T}_{2} and \mathcal{T}_{2} \geq \mathcal{T}_{1} together imply that \mathcal{T}_{2} = \mathcal{T}_{1}. In other words, these two topologies are equivalent, or they give rise to same basis.

Axiom 3: Transitivity holds because set inclusion/containment is transitive. If \mathcal{T}_{1} \geq \mathcal{T}_{2} and \mathcal{T}_{2} \geq \mathcal{T}_{3}, then clearly \mathcal{T}_{1} \geq \mathcal{T}_{2} \geq \mathcal{T}_{3}. In other words, there can be a chain of finer/coarser topologies for a given set.

Part 2:

TST: Under this definition of partial ordering of topologies, the partial ordering forms a lattice.

From axiom 3 proof, we know that there can exist a chain of finer/coarser topologies. So supremum and infimum can exist in a partial ordering of topologies. Hence, such a partial ordering forms a lattice.

QED.

Kindly let me know your comments and oblige about my above attempt.

Regards,

Nalin Pithwa

Solutions to Chapter 1: Topology, Hocking and Young

Exercise 1-4: Prove that the collection of all open half planes is a subbasis for the Euclidean topology of the plane.

Proof 1-4:

Note: In the Euclidean plane, we can take as a basis the collection of all interiors of squares.

Note also that a subcollection \mathcal{B} of all open sets of a topological space S is a SUBBASIS of S provided that the collection of all finite intersections of elements of $\mathcal{B}$ is a basis for S.

Clearly, from all open half planes (x > x_{i} and y > y_{i}), we can create a collection of all interior of squares.

Hence, the collection of all finite intersections of all open-half planes satisfies:

Axiom \mathcal{O_{2}}: the intersection of a finite intersection of open half planes is an open set. (interior of a square).

Axiom \mathcal{O_{1}}; (also). The union of any number of finite intersections of all open half planes is also open set (interior of a square).

Axiom O_{3}: (clearly). \phi and S are open.

QED.

Exercise 1-5:

Let S be any infinite set. Show that requiring every infinite subset of S to be open imposes the discrete topology on S.

Proof 1-5:

Case: S is countable. We neglect the subcase that a selected subcase is finite. (I am using Prof. Rudin’s definition of countable). The other subcase is that there exists a subset X_{1} \subset S, where X_{1} is countable. Let X_{1} be open. Hence, S-X_{1} is closed. But S-X_{1} is also countable. Hence, S-X_{1} is also open. Hence, there is no limit point. Hence, the topology is discrete, that is, there is no limit point.

Case: S is uncountable. Consider again a proper subset X_{1} \subset S; hence, X_{1} is open by imposition of hypothesis. Hence, S-X_{1} is closed. But, S-X_{1} \neq \phi and not finite also. Hence, S-X_{1} is infinite. Hence, S-X_{1} is open. Hence, there are no limit points. Hence, it is a discrete topology in this case also.

QED.

Your comments/observations are welcome !

Regards,

Nalin Pithwa.

The significance of mathematical logic: words of Alfred Tarski

Reference: Introduction to Logic, and to the Methodology of Deductive Sciences by Alfred Tarski, Oxford University Press, New York; available in Amazon India.

Just my two cents worth: Of course, Euclid introduced the deductive method about 2500 years back. Prof Tarski says there is more to it 🙂

(below I just present his motivational explanation in the preface, 1937 A.D)

In the opinion of many laymen mathematics is today already a dead science; after having reached an unusually high degree of development/sophistication, it has become petrified in rigid perfection. This is an entirely erroneous view of the situation; there are but few domains of scientific research which are passing through a phase of such intensive development at present as mathematics. Moreover, this development is extraordinarily manifold: mathematics is expanding in all possible directions, it is growing in height, in width, and in depth. It is growing in height, since, on the soil of its old theories which look back upon hundreds if not thousands of years of development, new problems appear again and again, and ever more perfect results are being achieved. It is growing in width, since its methhods permeate other branches of sciences, while its domain of investigation embraces increasingly more comprehensive ranges of phenomena and ever new theories are being included in the large circle of mathematical disciplines. And finally it is growing in depth, since its foundations become more and more firmly established, its methods perfected, and its principles stabilized.

It has been my intention in this book to give those readers who are interested in contemporary mathematics, without being actively concerned with it, at least a very general idea of that third line mathematical development, that is, its growth in depth. My aim has been to acquaint the reader with the most important concepts of a discipline which is known as mathematical logic, and which has been created for the purpose of a firmer and more profound establishment of the foundations of mathematics; this discipline, in spite of its brief existence of barely a century, has already attained a high degree of perfection and plays today a role in the totality of our knowledge that far transcends its originally intended boundaries. It has been my intention to show that the concepts of logic permeate the whole of mathematics, that they comprehend all specifically mathematical concepts as special cases, and that logical laws are constantly applied — be it consciously or unconsciously — in mathematical reasonings.

Cheers,

Nalin Pithwa.

The book is quite accessible with some moderate concentration 🙂

Solutions 1: Introduction to Logic, Alfred Tarski

Reference: Previous Blog; Introduction to Logic and to the Methodology of Deductive Sciences, by Alfred Tarski, chapter 1.

Exercises:

Question 1: Which among the following are sentential functions and which are designatory functions?

1a) x is divisible by 3: sentential function

1b) the sum of the numbers x and 2: designatory function.

1c) y^{2}-x^{2}: designatory function

1d) y^{2}=x^{2}: sentential function

1e) x+2<y+3: sentential function

1f) (x+3)-(y+5): designatory function

1g) the mother of x and z: designatory function

1h) x is the mother of z? : sentential function.

Question 2: Give examples of sentential and designatory functions from the field of geometry.

Answer 2: Designatory function: two parallel lines

Answer 2: Sentential function: Area of parallelogram with sides x and y is xy\sin{\theta} where \theta is the angle between the sides.

Question 3: The sentential functions which are encountered in arithmetic and which contain only one variable (which may, however, occur at several different places in the given sentential function) can be divided into three categories: (a) functions satisfied by every number (b) functions not satisfied by any number; (c) functions satisfied by some numbers, and not satisfied by others.

To which of these categories do the following sentential functions belong:

(i) x+2=5+x; category b.

(ii) x^{2}=49; category c.

(iii) (y+2)(y-2)<y^{2}; category a.

(iv) y+24>36; category c.

(v) z=0 or z<0 or z>0; category a.

(vi) z+24>z+36? category b.

Question 4: Give examples of universal, absolutely existential and conditionally existential theorems from the fields of arithmetic and geometry.

Answer 4:

Universal existential theorem: arithmetic: commutative law: x+y=y+x

Universal existential theorem: geometry: Parallel lines do not meet.

Absolutely existential theorem: arithmetic: there are numbers x and y such that x<y+1

Absolutely existential theorem: geometry: there are three points which can form a triangle.

Conditionally existential theorem: arithmetic: For any numbers x and y, there is a number z such that x=y+z

Conditionally existential theorem: geometry: For any two lines which meet, there is an angle bisector.

Question 5: By writing quantifiers containing the variables “x” and “y” in front of the sentential function:

x>y

it is possible to obtain various sentences from it, for instance:

(i) for any numbers x and y, x>y; (not always true)

(ii) for any number x, there exists a number y such that x>y; (always true)

(iii) there is a number y such that, for any number x, x>y. (not always true)

Formulate them all (there are six altogether) and determine which of them are true.

The other three possibilities are as follows:

(iv) given x, there is no y such that x>y; false.

(v) given y, there is no x such that x>y; false.

(vi) there is no x or y such that x>y; false.

Question 6: Do the same as in Exercise 5 for the following existential functions:

x+y^{2}>1 and “x is the father of y”.

(assuming that the variables ‘x” and “y” in the latter stand for names of human beings.)

part 1: x+y^{2}>1

1a: for any numbers x and y, x+y^{2}>1 (not always true)

1b: for any number x, there exists a number y such that x+y^{2}>1 (always true)

1c: there is a number y such that, for any number x, x+y^{2}>1 (always true)

1d: given x, there is no y such that x+y^{2}>1 (always false)

1e: given y, there is no x such that x+y^{2}>1 (always false)

1f: there is no x or y such that x=y^{2}>1; always false.

Part 2: x is the father of y:

2a: for any x and y, x is the father of y (not always true)

2b: for any x, there exists a y such that x is the father of y. (not always true)

2c: there is a y such that for any x, x is the father of y. (not always true).

2d: given x, there is no y such that “x is the father of y” (not always true).

2e: given y, there is no x such that “x is the father of y” (false).

2f: there is no x or y such that “x is the father of y” (false).

Question 7: State a question of every day language that has the same meaning as:

for every x, if x is a dog, then x has a good sense of smell.

Answer 7: Every dog has a good sense of smell.

Question 8: Replace the sentence: “some snakes are poisonous” by one which has the same meaning but is formulated with the help of quantifiers and variables.

Answer 8; There exist some snakes which are poisonous.

Question 9: Differentiate, in the following expressions, between the free and bound variables:

9a: x is divisible by y: y is free, x is bound.

9b: for any x, x-y=x+(-y) ; both x and y are free.

9c: if x<y, then there is a number z such that x<y and y<z; x and y are free, z is bound.

9d: for any number y, if y>0. then, for any number z such that x=y.z; z is bound; x is bound, y is free.

9e: if x=y^{2} and y>0, then, for any number z, x>-z^{2}; x and y are bound; z is not bound.

PS: I am skipping two, three questions in the exercise.

Your comments are welcome!

I hope to continue this logic blog albeit quite slow.

Cheers,

Nalin Pithwa.

Tutorial Problems I: Topology: Hocking and Young

Reference: Topoology by Hocking and Young, Dover Publications, Inc., NY. Available in Amazon India.

Exercises 1-1:

Show that if S is a set with the discrete topology and f: S \rightarrow T is any transformation of S into a topologized set T, then f is continuous.

Solution 1-1:

Definition A: The set S has a topology (or is topologized) provided that, for every point p in S and every subset X of S, the question : “is p a limit point of X?” can be answered.

Definition B: A topology is said to be a discrete topology when we assume that for no point p in S, and every subset X of S: the answer to the question: “is p a limit point of X?” is NO.

Definition C: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

So, the claim is vacuously true. QED.

Exercises 1-2:

A real-valued function y=f(x) defined on an interval [a,b] is continuous provided that if a \leq x_{0} \leq b and \epsilon >0, then there is a number \delta>0 such that if $|x-x_{0}|<\delta$, where x \in [a,b], then |f(x)-f(x_{0})|<\epsilon. Show that this is equivalent to our definition, using definition 1-1.

Solution 1-2:

Definition 1-1: The real number p is a limit point of a set X of real numbers provided that for every positive number \epsilon, there is an element x of the set X such that $0<|p-x|<\epsilon$.

Definition C: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

Part 1: Let us assume that given function f is continuous as per definition given just above.

Then, as p is a limit point of X: it means: For any \delta>0, there exists a real number p such that there is an element x \in X such that |p-x|<\delta..

So, also, by definition C, f(p) is a limit point or a point of f(X); this means the following: if f(p) is a point of f(X), there exists some x_{0} \in X such that $f(x_{0} \in f(X)$, and so quite clearly in this case p=x_{0} so that |p-x_{0}|=|x_{0}-x_{0}|<\delta, as \delta is positive.

On the other hand, if f(p) is a limit point of f(X), as per the above definition of continuity, then also for any \epsilon>0, there exists a point y \in f(X) such that |y-f(p)|<\epsilon. So, in this case also the claim is true.

We have proved Part 1. QED.

Now, part II: We assume the definition of continuity given in the problem statement is true. From here, we got to prove definition C as the basic definition given by the authors.

But this is quite obvious as in this case p=x_{0}.

We have proved Part II. QED.

Thus, the two definitions are equivalent.

Cheers,

Nalin Pithwa

Exercises 1: Alfred Tarski, Introduction to Logic

I. Which among the following expressions are sentential functions, and which are designatory functions:

a) x is divisible by 3

b) the sum of the numbers x and 2

c) x^{2}-z^{2}

d) y^{2}=z^{2}

e) x+2< y+3

f) (x+3) - (y+5)

g) the mother of x and z

h) x is the mother of z?

Problem 2: Give examples of sentential and designatory functions from the field of geometry.

Problem 3: The sentential functions which are encountered in arithmetic and which contain only one variable (which may, however, occur at several different places in the given sentential function) can be divided into three categories : (1) functions satisfied by every number; (ii) functions not satisfied by any number; (iii) functions satisfied by some numbers, and not satisfied by others.

To which of these categories do the following sentential functions belong:

(a) x+2=5+x

(b) x^{2}=49

(c) (y+2).(y-2)<y^{2}

(d) y+24>36

(e) z=0 or z<0 or z>0

(f) z+24>z+36?

Problem 4:Give examples of universal, absolutely existential and conditionally existential theorems from the fields of arithmetic and geometry.

Problem 5: By writing quantifiers containing the variables “x” and “y” in front of the sentential function: x>y it is possible to obtain various sentences from it, for instance:

for any numbers x and y, x>y;

for any number x, there exists a number y such that x>y;

there is a number y such that, for any number x, x>y.

Formulate them all (there are six altogether) and determine which of them are true.

Problem 6: Do the same as in problem 5 for the following sentential functions:

x+y^{2}>1 and ” x is the father of y.”

(assuming that the variables x and y in the latter stand for names of human beings.)

Problem 7: State a sentence of every day language that has the meaning as:

For every x, if x is a dog, then x has a good sense of smell.

And, your sentence must not contain any quantifier or variables.

Problem 8:

Replace the following sentence: “some snakes are poisonous” by one which has the same meaning but is formulated with the help of quantifiers and variables.

Problem 9:

Differentiate, in the following expressions, between the free and bound variables:

(a) x is divisible by y.

(b) for any x, x-y = x +(-y)

(c) if x<y, then there is a number z such that <y and y<z;

(d) for any number y, if y>0, then there is a number z such that x=y.z

(e) if x=y^{2} and y>0, then for any number z, x>-z^{2};

(f) if there exists a number y such that x>y^{2}, then, for any number z, x>-z^{2}.

Formulate the above expressions by replacing the quantifiers by the symbols introduced in Section 4.

Problem 10*: If, in the sentential function, (e) of the preceding exercise, we replace the variable “z” in both places by “y”, we obtain an expression in which “y” occurs in some places as a free and in others as a bound variable; in what places and y?

(In view of some difficulties in operating with expressions in which the same variable occurs both bound and free, some logicians prefer to avoid the use of such expressions altogether and not to treat them as sentential functions.)

Problem 11*: Try to state quite generally under which conditions a variable occurs at a certain place of a given sentential function as a free or as a bound variable.

Problem 12: Which numbers satisfy the sentential function: there is a number y such that x=y^{2}, and which satisfy: there is a number y such that x.y=1?

Cheers,

Nalin Pithwa

If you are pursuing higher math

Experience seems to show that the student usually finds a new theory difficult to grasp at a first reading. He needs to return to it several times before he becomes really familiar with it and can distinguish for himself which are the essential ideas and which results are of minor importance, and only then will he be able to apply it intelligently.

— quoted by Jean Dieudonne, in his preface to Foundations of Modern Analysis. (Academic Press, NY and London, 1969).