# Category: Topology

# An example: continuous map which is not homeomorphic

Reference: Topology by Hocking and Young. Dover Publications.

Let us present an example of a continuous mapping (one-to-one) which is not a homeomorphism:

Let S be the set of all non-negative real numbers with their usual metric topology, and let T be the unit circle in its metric topology. Consider $f: S \rightarrow T$. For each x in S, let $f(x) = (1, \frac{2\pi x^{2}}{(1+x^{2})})$, a point in polar coordinates on T.

It is easily shown that f is continuous and one-to-one.

But the set of all x in S such that $x<1$ is open in S while its image is not open in T. Hence, f is not interior. (meaning that: A transformation $f: S \rightarrow T$ of the space S into the space T is said to be interior if f is continuous and and if the image of every open subset of S is open in T) , and is not a homeomorphism (because of the following theorem: A necessary and sufficient condition that the one-to-one continuous map $f: S \rightarrow T$ of the space S onto the space S be a homeomorphism is that f be interior).

Regards,

Nalin Pithwa.

# 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 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 , and such that 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 os a collection of connected subsets of S, each of which intersects C. Then, is connected.

Corollary of above: For each n, is connected.

Theorem 4:

Every continuous image of a connected space is connected.

Lemma 3: For , the complement of the origin in is connected.

Theorem 5: For each , is connected.

Theorem 6: If both and 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 in 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 is continuous provided that if p is a limit point of a subset X of S, then is a limit point or a point of .

Definition 2: We may also state the continuity requirement on f as follows: if p is a limit point of , then is a point of .

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

Theorem 2: A real-valued function defined on an interval is continuous provided that if and , then there is a number such that if , x in , then . (NB: this is same as definition 1 above).

Theorem 3: Let 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 is open in S.

Theorem 4:

A necessary and sufficient condition that the transformation 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 , then there is an open set, U in S containing x and such that lies in V.

Theorem 5:

A one-to-one transformation of a space S onto a space T is a homeomorphism, if and only if both f and are continuous.

Theorem 6:

Let be a transformation of the metric space M with metric d into the metric space N with metric . A necessary and sufficient condition that f be continuous is that if is any positive number and x is a point of M, then there is a number such that if , then .

Theorem 7:

A necessary and sufficient condition that the one-to-one mapping (that is, a continuous transformation) of the space S onto the space T be a homeomorphism is that f is interior. (NB: A transformation 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 of a space S onto a space T is a homeomorphism if and only if both f and 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 there exists a point such that .

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

But, the above two definitions when combined mean the following: at least for the case of a Euclidean space: A real valued function defined on an interval is continuous provided that if and , then there is a number such that if , x in , then .

The above sub-case settles the proof for the Euclidean space .

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

Theorem 1-6: Let 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 is open in S.

Next, theorem 1-7: A necesssary and sufficient condition that the transformation 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 , then there is an open set, U in S containing x such that 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 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 and be two collections of subsets of a set S, both satisfying the three defining axioms , and of a topology. Let S have two topologies. We will say that the topology determined by is a finer topology than the topology determined by if every set is a union of sets , that is, each is open in the topology. We will denote this situation with the symbol .

Axiom 1: reflexive clearly holds because as set inclusion/containment is reflexive.

Axiom 2: antisymmetric also holds true because clearly and together imply that . 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 and , then clearly . 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 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 ( and ), we can create a collection of all interior of squares.

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

Axiom : the intersection of a finite intersection of open half planes is an open set. (interior of a square).

Axiom ; (also). The union of any number of finite intersections of all open half planes is also open set (interior of a square).

Axiom : (clearly). 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 , where is countable. Let be open. Hence, is closed. But is also countable. Hence, 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 ; hence, is open by imposition of hypothesis. Hence, is closed. But, and not finite also. Hence, is infinite. Hence, 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.

# 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 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 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 defined on an interval [a,b] is continuous provided that if and , then there is a number such that if $|x-x_{0}|<\delta$, where , then . 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 , there is an element x of the set X such that $0<|p-x|<\epsilon$.

Definition C: A transformation 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 , there exists a real number p such that there is an element such that ..

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 such that $f(x_{0} \in f(X)$, and so quite clearly in this case so that , as 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 , there exists a point such that . 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 .

We have proved Part II. QED.

Thus, the two definitions are equivalent.

Cheers,

Nalin Pithwa

# VI. Countable sets: My notes.

Reference:

- Introduction to Topology and Modern Analysis by G. F. Simmons, Tata McGraw Hill Publications, India.
- Introductory Real Analysis — Kolmogorov and Fomin, Dover Publications, N.Y.(to some extent, I have blogged this topic based on this text earlier. Still, it deserves a mention/revision again).
- Topology : James Munkres.

The subject of this section and the next — ** infinite cardinal numbers **— lies at the very foundation of modern mathematics. It is a vital instrument in the day to day work of many mathematicians, and we shall make extensive use of it ourselves (in our beginning studying of math ! :-)). This theory, which was created by the German mathematician Georg Cantor, also has great aethetic appeal, for it begins with ideas of extreme simplicity and develops through natural stages into an elaborate and beautiful structure of thought. In the course of our discussion, we shall answer questions which no one before Cantor’s time thought to ask, and we shall ask a question which no one can answer to this day…(perhaps !:-))

Without further ado, we can say that * cardinal numbers are those used in counting, such as the positive integers (or natural numbers) 1, 2, 3, …familiar to us all. *But, there is much more to the story than this.

The act of counting is undoubtedly one of the oldest of human activities. Men probably learned to count in a crude way at about the same time as they began to develop articulate speech. The earliest men who lived in communities and domesticated animals must have found it necessary to record the number of goats in the village herd by means of a pile of stones or some similar device. If the herd was counted in each night by removing one stone from the pile for each goat accounted for, then stones left over would have indicated strays, and herdsmen would have gone out to search for them. Names for numbers and symbols for them like our 1, 2, 3, …would have been superfluous. *The simple yet profound idea of a one-to-one correspondence between the stones and the goats would have fully met the needs of the situation.*

In a manner of speaking, we ourselves use the infinite set

of all positive integers as “pile of stones.” We carry this set around with us as part of our intellectual equipment. Whenever we want to count a set, say, a stack of dollar bills, we start through the set N and tally off one bill against each positive integer as we come to it. The last number we reach, corresponding to the last bill, is what we call the number of bills in the stack. If this last number happens to be 10, then “10” is our symbol for the number of bills in the stack, as it also is for the number of our fingers, and for the number of our toes, and for the number of elements which can be put into one-to-one correspondence with the finite set . Our procedure is slightly more sophisticated than that of the primitive savage man. We have the symbols 1, 2, 3, …for the numbers which arise in counting; we can record them for future use, and communicate them to other people, and manipulate them by the operations of arithmetic. But the underlying idea, that of the one-to-one correspondence, remains the same for us as it probably was for him.

The positive integers are adequate for our purpose of counting any non-empty finite set, and since outside of mathematics all sets appear to of this kind, they suffice for all non-mathematical counting. But in the world of mathematics, we are obliged to consider many infinite sets, such as the set of positive integers itself, the set of all integers, the set of all rational numbers, the set of all real numbers, the set of all points in a plane, and so on. It is often important to be able to count such sets, and it was Cantor’s idea to do this, and to develop a theory of infinite cardinal numbers, by means of one-to-one correspondence.

In comparing the sizes of two sets, the basic concept is that of numerical equivalence as defined in the previous section. We recall that two non-empty sets are numerically equivalent if there exists a one-to-one mapping of a set onto the other, or — and this amounts to the same thing — if there can be found a one-to-one correspondence between them. To say that two non-empty finite sets are numerically equivalent is of course to say that they have the same number of elements in the ordinary sense. If we count one of them, we simply establish a one-to-one correspondence between its elements and a set of positive integers of the form and we then say that n is the number of elements possessed by both, or the cardinal number of both. ** The positive integers are the finite cardinal numbers.** We encounter many surprises as we follow Cantor and consider numerical equivalences for infinite sets.

The set of all positive integers is obviously “larger” than the set of all even positive integers, for it contains this set as a proper subset. It appears on the surface that N has “more” elements. But it is very important to avoid jumping to conclusions when dealing with infinite sets, and we must remember that our criterion in these matters is whether there exists a one-to-one correspondence between the sets (not whether one set is a proper subset or not of the other) . As a matter of fact, consider the “pairing”

serves to establish a one-to-one correspondence between these sets, in which each positive integer in the upper row is matched with the even positive integer (its double) directly below it, and these two sets must therefore be regarded as having the same number of elements. This is a very remarkable circumstance, for it seems to contradict our intuition and yet is based only on solid common sense. We shall see below, in Problems 6 and 7-4, that every infinite set is numerically equivalent to a proper subset of itself. Since this property is clearly not possessed by any finite set, some writers even use it as the definition of an infinite set.

In much the same way as above, we can show that N is numerically equivalent to the set of all even integers:

Here, our device is start with zero and follow each even positive integer as we come to it by its negative. Similarly, N is numerically equivalent to the set of all integers:

It is of considerable interest historical interest to note that Galileo had observed in the early seventeenth century that there are precisely as many perfect squares (1,4,9,16,25, etc.) among the positive integers as there are positive integers altogether. This is clear from the “pairing”:

It struck him as very strange that this should be true, considering how sparsely strewn the squares are among all the positive integers. But, the time appears not to have been ripe for the exploration of this phenomenon, or perhaps he had other things on his mind; in any case, he did not follow up his idea.

These examples should make it clear that all that is really necessary in showing that an infinite set X is numerically equivalent to N is that we be able to list the elements of X, with a first, a second, a third, and so on, in such a way that it is completed exhausted by this counting off of its elements. It is for this reason that any infinite set which is numerically equivalent to N is said to be * countably infinite.* (Prof. Walter Rudin also, in his classic on mathematical analysis, considers a countable set to be either finite or countably infinite. ) We say that a set is countable it is non-empty and finite (in which case it can obviously be counted) or if it is countably infinite.

One of Cantor’s earliest discoveries in his study of infinite sets was that the set of all positive rational numbers (which is very large : it contains N and a great many other numbers besides) is actually countable. We cannot list the positive rational numbers in order of size, as we can the positive integers, beginning with the smallest, then the next smallest, and so on, for there is no smallest, and between any two there are infinitely many others. We must find some other way of counting them, and following Cantor, we arrange them not not in order of size, but according to the size of the sum of numerator and denominator. We begin with all positive rationals whose numerator and denominator sum add up to 2: there is only one . Next, we list (with increasing numerators) all those for which this sum is 3: . Next, all those for which the sum is 4: . Next, all those for which this sum is 5: . Next, all those for which this sum is 6: . And, so on. If we now list all these together from the beginning, omitting those already listed when we come to them, we get a sequence like:

which contains each positive rational number once and only once. (There is a nice schematic representation of this : Cantor’s diagonalization process; please google it). In this figure, the first row contains all positive rationals with numerator 1, the second all with numerator 2, and so on. Our listing amounts to traversing this array of numbers in a zig-zag manner (again, please google), where of course, all those numbers already encountered are left out.

It is high time that we christened the infinite cardinal number we have been discussing, and for this purpose, we use the first letter of the Hebrew alphabet, . We say that is the number of elements in any countably infinite set. Our complete list of cardinal numbers so far is

.

We expand this list in the next section.

Suppose now that m and n are two cardinal numbers (finite or infinite). The statement that m is less than n (written m < n) is defined to mean the following: if X and Y are sets with m and n elements, then (i) there exists a one-to-one mapping of X into Y, and (ii) there does not exist a mapping of X onto Y. Using this concept, it is easy to relate our cardinal numbers to one another by means of:

.

With respect to the finite cardinal numbers, this ordering corresponds to their usual ordering as real numbers.

Regards,

Nalin Pithwa

# V. Exercises: Partitions and Equivalence Relations

Reference: Topology and Modern Analysis, G. F. Simmons, Tata McGraw Hill Publications, India.

- Let be an arbitrary mapping. Define a relation in X as follows: means that . Prove that this is an equivalence relation and describe the equivalent sets.

Proof : HW. It is easy. Try it. ðŸ™‚

2. In the set of real numbers, let means that is an integer. Prove that this is an equivalence relation and describe the equivalence sets.

Proof: HW. It is easy. Try it. ðŸ™‚

3. Let I be the set of all integers, and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m — symbolized by — if a-b is exactly divisible by m, that is, if is an integral multiple of m. Show that this is an equivalence relation, describe the equivalence sets, and state the number of distinct equivalence sets.

Proof: HW. It is easy. Try it. ðŸ™‚

4. Decide which one of the three properties of reflexivity, symmetry and transitivity are true for each of the following relations in the set of all positive integers: , , . Are any of these equivalence relations?

Proof. HW. It is easy. Try it. ðŸ™‚

5. Give an example of a relation which is (a) reflexive, but not symmetric or transitive. (b) symmetric but not reflexive or transitive. (c) transitive but not reflexive or symmetric (d) reflexive and symmetric but not transitive (e) reflexive and transitive but not symmetric. (f) symmetric and transitive but not reflexive.

Solutions. (i) You can try to Google (ii) Consider complex numbers (iii) there are many examples given in the classic text “Discrete Mathematics” by Rosen.

6) Let X be a non-empty set and a relation in X. The following purports to be a proof of the statement that if this relation is symmetric and transitive, then it is necessarily reflexive: ; and ; therefore, for every x. In view of problem 5f above, this cannot be a valid proof. What is the flaw in the reasoning? ðŸ™‚

7) Let X be a non-empty set. A relation in X is called circular if and , and triangular if . Prove that a relation in X is an equivalence relation if and only if it is reflexive and circular if and only if it is reflexive and triangular.

HW: Try it please. Let me know if you need any help.

Regards,

Nalin Pithwa.

PS: There are more examples on this topic of relations in Abstract Algebra of I. N. Herstein and Discrete Mathematics by Rosen.