## Read an Excerpt

#### Topology

**By John G. Hocking, Gail S. Young**

**Dover Publications, Inc.**

**Copyright © 1961 John G. Hocking and Gail S. Young**

All rights reserved.

ISBN: 978-0-486-14109-1

All rights reserved.

ISBN: 978-0-486-14109-1

CHAPTER 1

**TOPOLOGICAL SPACES AND FUNCTIONS**

**1-1 Introduction**. Topology may be considered as an abstract study of the limit-point concept. As such, it stems in part from a recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. The very definition of a continuous function is an example of this dependence. Another example is the precise meaning of the connectedness of a geometric figure. To exaggerate, one might view topology as the complement of modern algebra in that together they cover the two fundamental types of operations found in mathematics.

In applying the unifying principle of abstraction, we study concrete examples and try to isolate the basic properties upon which the interesting phenomena depend. In the final analysis, of course, the determination of the "correct" properties to be abstracted is largely an experimental process. For instance, although the limit of a sequence of real numbers is a widely used idea, experience has shown that a more basic concept is that of a limit point of a *set* of real numbers.

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

As an example, let *X* consist of all real numbers of the two forms 1/*n* and (*n* - 1)*/n,* where *n* is an integer greater than 2. Then 0 and 1 are the only limit points of *X.* Thus a limit point of a set need not belong to that set. On the other hand, every real number is a limit point of the set of all rational numbers, indicating that a set may have limit points belonging to itself.

Some terminology is needed before we pursue this abstraction further. Let *S* be any set of elements. These may be such mathematical entities as points in the Euclidean plane, curves in a given class, infinite sequences of real numbers, elements of an algebraic group, etc., but in general we take *S* to be an abstract undefined set. To reflect the geometric content of topology, we refer to the elements of *S* by the generic name *point.* We may now name our fundamental structure.

Definition 1-2. 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.

This definition is so extremely general as to be almost useless in practice. There is nothing in it to impose certain desirable properties upon the limit-point relation (more on this point shortly), and also nothing in it indicates the means whereby the pertinent question can be answered. An economical method of accomplishing the latter is to adopt some rule or test whose application will answer the question in every case. For the set of real numbers, **Definition 1-1** serves this purpose and hence defines a topology for the real numbers. [The use of the word *topology* here differs from its use as the name of a subject. Loosely speaking, topology (the subject) is the study of topologies (as in **Definition 1-2**).]

A set *S* may be assigned many different topologies, but there are two extremes. For the first, we always answer the question in **Definition 1-2** in the affirmative; that is, every point is a limit point of every subset. This yields a worthless topology: there are simply too many limit points! For the other extreme, we assume that the answer is always "no," that is, no point is a limit point of any set. The resulting topology is called *the discrete topology* for *S.* The very fact that it is dignified with a name would indicate that this extreme is not quite so useless as the first.

Those factors that dictate the choice of a topology for a given set *S* should become more apparent as we progress. In many cases, a "natural" topology exists, a topology agreeing with our intuitive idea of what a limit point should be. **Definition 1-1** furnishes such a topology for the real numbers, for instance. In general, however, we require only a structure within the set *S* which will define *limit point* in a simple manner and in such a way that certain basic relations concerning limit points are maintained. To illustrate this latter requirement, it is intuitively evident that if *p* is a limit point of a subset *X* and *X* is contained in another subset *Y,* then we would want *p* to be also a limit point of F. There are many such structures one may impose upon a set and we will develop the more commonly used topologies in this chapter. Before doing this, however, we continue our preliminary discussion with a few general remarks upon the aims and tools of topology.

The study of topologized sets (or any other abstract system) involves two broad and interrelated questions. The first of these concerns the investigation and classification of the various concrete realizations, or models, which we may encounter. This entails the recognition of equivalent models, as is done for isomorphic groups or congruent geometric figures, for example. In turn, this equivalence of models is usually defined in terms of a one-to-one reversible transformation of one model onto another. This equivalence transformation is so chosen as to leave invariant the fundamental properties of the models. As examples, we have the rigid motions in geometry, the isomorphisms in group theory, etc.

One of the first to perceive the importance of these underlying transformations was Felix Klein. In his famous Erlanger Program (1870), he characterized the various geometries in terms of these basic transformations. For instance, we may define Euclidean geometry as thestudy of those properties of geometric figures that are invariant under the group of rigid motions.

*Insofar as topology is an abstract form of geometry and fits into the Klein Erlanger Program, its basic transformations are the homeomorphisms* (which we will define shortly).

The second broad question in studying an abstract system such as our topologized sets involves consideration of transformations more general than the one-to-one equivalence transformation. The requirement that the transformation be one-to-one and reversible is dropped and we retain only the requirement that the basic structure is to be preserved. The homomorphisms in group theory illustrate this situation. In topology, the corresponding transformations are those that preserve limit points. Such a transformation is said to be *continuous* and is a true generalization of the continuous functions used in analysis. It follows that second aspect of topology finds many applications in function theory.

Since we are to be dealing with very general sets, we must give precise meaning to the word *transformation.*

Definition 1-3. Given two sets *X* and *Y,* a *transformation* (also called a *function* or a *mapping*) *f:X -> Y* of *X* into *Y* is a triple (*X**Y,**G*), where *G* itself is a collection of ordered pairs (*x, y*), the first element of each pair being an element of *X,* and the second an element of *Y,* with the condition that each element of *X* appears as the first element of exactly one pair in *G.*

If each element of *Y* appears as the second element of some pair in *G,* then the transformation *f* is said to be *onto.*

If each element of *Y* which appears at all, appears as the second element of exactly one pair in *G,* then *f* is said to be *one-to-one.* Note that a transformation can be onto without being one-to-one and conversely.

As an aid in understanding **Definition 1-3**, consider the equation *y = x2, x* a real number. We may take *X* to be the set of all real numbers and then the collection *G* is the set of pairs (*x, x*2). From this alone, we cannot determine the set *Y,* however. Certainly *Y* must contain all nonnegative real numbers since each such number appears as the second element of at least one pair (*x, x2*). Taking *Y* to be just the set of nonnegative reals will cause *f* to be onto. But if *Y* is all real numbers, or all reals greater than -7, or any other set containing the nonnegative reals as a proper subset, the transformation is not onto. With each new choice of *Y,* we change the triple and hence the transformation.

Continuing with the same example, we could assume that *X* is the set of nonnegative reals also. Then the transformation is one-to-one, as is easily seen. Depending upon the choice of *Y,* the transformation may or may not be onto, of course. Thus we see that we have stated explicitly the conditions usually left implicit in defining a function in elementary analysis. The reader will find that the seemingly pedantic distinctions made here are really quite necessary.

If *f:X -> Y* is a transformation of *X* into *Y* and *x* is an element of the set *X,* then we let *f* (*x*) denote the second element of the pair in *G* whose first element is *x.* That is, *f* (*x*) is the "functional value" in *Y* of the point *x.* Similarly, if *Z* is a subset of *X,* then *f* (*Z*) denotes that subset of *Y* composed of all points *f* (*z*), where *z* is a point in *Z.* If *y* is a point of *Y,* then by *f*-1(*y*) is meant the set of all points *x* in *X* for which *f* (*x*) = *y;* and if *W* is a subset of *Y,* then *f*-1(*W*) is the set-theoretic union of the sets *f-1w* in *W.* Note that *f*-1 can be used as a symbol to denote the triple (*Y, X, G'*), where *G'* consists of all pairs (*y,**x*) that are reversals of pairs in *G.* But *f-1**is a transformation only if f is both one-to-one and onto.* If *A* is a subset of X and if *f:X -> Y*, then *f* may be restricted to *A* to yield a transformation denoted by *f|A: A -> Y*, and called the *restriction* of *f* to *A.*

We can now define the transformations that underlie the study of topology. Let *S* and *T* be topologized sets. A *homeomorphism* of *S* onto *T* is a one-to-one transformation *F:S -> 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). This last condition means that a homeomorphism preserves limit points, a condition that is certainly natural enough if we expect to study limit points. Note that since a homeomorphism *f* is both one-to- one and onto, its inverse *f1* is also a transformation. Furthermore the "if and only if" condition implies that *f-1* is also a homeomorphism f-1: T -> S.

One might consider the homeomorphism as the analogue of an isomorphism in algebra, or a conformal mapping in analysis, or a rigid motion in geometry. The less restricted class of continuous transformations mentioned earlier are then analogous to the homomorphisms in algebra, or analytic functions in analysis, or projections onto a lower-dimensional subspace in geometry. A transformation f:S -> 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*).

By introducing a new symbol, we can express continuity more concisely. If *X* is a subset of the topologized set *S,* we let [bar.X] denote the set-theoretic union of *X* and all its limit points and call X the *closure* of X. The continuity requirement on / then may be expressed by assuming that if *p* is a point of [bar.X], then *f(p)* is a point of [bar.*f(X]).*

Exercise 1-1. Show that if *S* is a set with the discrete topology and *f:S -> T* is any transformation of *S* into a topologized set *T,* then *f* is continuous.

Exercise 1-2. A real-valued function *y = f(x)* defined on an interval [*a, b*] is continuous provided that if *a* ≤ *x0* ≤ *b* and ε > 0, then there is a number δ > 0 such that if |*x - x0*| < δ, x in [*a, b*], then |*f(x) - f(x0)*| Show that this is equivalent to our definition, using **Definition 1-1.**

**1-2 Topological spaces** . In attempting to formulate a rule to use in answering the pertinent question in **Definition 1-2**, we should be guided by the properties of limit points and their relationships as found in analysis, where this abstraction began. For instance, we would not welcome a situation in which a point *p* is a limit point of the set of limit points of a set X and yet *p* is not a limit point of *X* itself. The structure we present first to accomplish our aims is widely adopted.

Consider a set *S.* Let {0α} be a collection of subsets of *S,* called *open sets,* satisfying the following axioms:

*O*1. The union of any number of open sets is an open set.

*O*2. The intersection of a finite number of open sets is an open set.

*O*3. Both *S* and the empty set θ are open.

With such a collection {Oα} we now determine the limit points of a subset as follows. A point *p* is a *limit point of a subset X* of *S* provided that every open set containing *p* also contains a point of *X* distinct from *p.* This definition yields a topology for *S* and, with such a topology, *S* is called a *topological space.*

Note that not every set with a topology is a topological space. If *S* is a topologized set, then for *S* to be a topological space, it must be possible to obtain the given topology by selecting certain subsets of *S* as open sets satisfying *O1, O*2, and *O*3 and to recover the given limit-point relations, using these open sets.

We now suppose that we have a topological space *S* with open sets {*O*α}. We define a subset *X* of *S* to be *closed* if *S - X* is open.

Theorem 1-1. If *X* is any subset of *S,* then *X* is closed if and only if *X = [bar.X].*

*Proof:* Suppose *X = [bar.X].* Then no point of *S - X* is a point or a limit point of *X.* About each point *p* in *S X,* then, there is an open set *0P* containing no point of *X.* By Axiom *o*1, the union of all the sets *0P, p* in *S - X,* is an open set. Clearly this union is *S - X.*

Conversely, if *X* is closed, then *S - X* is open. If *p* is any point of *S* - X, then *S* - X itself is an open set containing *p* but no point of X. Hence no point of *S - X* can be a limit point of X.

Theorem 1-2. The closed subsets {*C*α} of a topological space *S* satisfy the following properties:

*C*1. The intersection of any number of closed sets is closed.

*C*2. The union of a finite number of closed sets is closed.

*C*3. Both *S* and the empty set θ are closed.

*(Continues...)*

Excerpted fromTopologybyJohn G. Hocking, Gail S. Young. Copyright © 1961 John G. Hocking and Gail S. Young. Excerpted by permission of Dover Publications, Inc..

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.