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    Topology
    Topology
    Topology
    Ebook671 pages9 hours

    Topology

    By John G. Hocking and Gail S. Young

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    About this ebook

    "As textbook and reference work, this is a valuable addition to the topological literature." — Mathematical Reviews
    Designed as a text for a one-year first course in topology, this authoritative volume offers an excellent general treatment of the main ideas of topology. It includes a large number and variety of topics from classical topology as well as newer areas of research activity.
    There are four set-theoretic chapters, followed by four primarily algebraic chapters. Chapter I covers the fundamentals of topological and metrical spaces, mappings, compactness, product spaces, the Tychonoff theorem, function spaces, uniform continuity and uniform spaces. The next two chapters are devoted to topics in point-set topology: various separation axioms, continua in Hausdorff spaces, real-valued functions, and more Chapter IV is on homotopy theory. Chapter V covers basic material on geometric and abstract simplicial complexes and their subdivisions. Chapter VI is devoted to simplicial homology theory, Chapter VII covers various topics in algebraic topology, including relative homology, exact sequences, the Mayer-Vietoris sequence, and more. Finally, Chapter VIII discusses Cech homology.
    There are a large number of illuminating examples, counter-examples and problems, both those which test the understanding and those which deepen it. The authors have also made a special effort to make this an "open-ended" book, i.e while many topics are covered, there is much beyond the confines of this book. In many instances they have attempted to show the direction in which further material may be found.
    Topology is so fundamental, its influence is apparent in almost every other branch of mathematics, as well as such fields as symbolic logic, mechanics, geography, network theory, and even psychology. This well-written text offers a clear and careful exposition of this increasingly important discipline.

    LanguageEnglish
    PublisherDover Publications
    Release dateMay 23, 2012
    ISBN9780486141091
    Topology

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      Delightful text emphasizing point-set and the significance of limit sets. Much more readable than the elegant, yet rarified Kolmolgorov.

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    Topology - John G. Hocking

    INDEX

    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 , there is an element x of the set X

    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 the study 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 mappingof 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, x²). 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.

    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 , then f may be restricted to A 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 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 f-1 is also a transformation. Furthermore the if and only if condition implies that f-1

    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

    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

    X

    , then f(p) is a point of

    f(X)

    .

    EXERCISE 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.

    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 < δ, x in [a, b], then 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:

    O1. The union of any number of open sets is an open set.

    O2. The intersection of a finite number of open sets is an open set.

    O3. Both S 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, O2, and O3 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 {Oα} . 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 =

    X

    .

    Proof: Suppose X =

    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 o1, 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

    THEOREM 1-2. The closed subsets {Cα} of a topological space S satisfy the following properties:

    C1. The intersection of any number of closed sets is closed.

    C2. The union of a finite number of closed sets is closed.

    C3. Both S are closed.

    Proof: Let {C′β} be any subcollection of {Cα}. For each set C′β S — C′β is open. Hence by Oby de Morgan's law. This proves C1.

    Let C′1, … , C′n be any finite subcollection of {Cα}. Then each S — C′, is open and by Ois closed, proving (C2. Property C3 follows immediately from O

    This result is actually a theorem in pure set theory, not in topology. It depends only upon de Morgan's law, which asserts that if S is any set and {Xα is any collection of subsets of S. A proof of this property is rather easy and is available in any treatise on the theory of sets. For example, see Fraenkel [8].

    We might point out the obvious formal duality between Properties C1, C2, C3 and Axioms O1, O2, O3. One may always pass from true statements about open sets to true statements about closed sets by interchanging open set with closed set and union with intersection throughout. This would be much too formal an approach, however, and defining a topological space via its closed sets lacks certain advantages which we will bring out in the next section.

    1-3 Basis and subbasis of a topology. One justification for considering open sets is a desire to reduce the number of subsets that one must study in order to define a topology. If c is the cardinal number of the set of real numbers, for example, then the set of all subsets of the real numbers has cardinal number 2c, a larger infinity than c. To decide set by set and point by point which points are to be limit points of which sets would require c · 2c = 2c decisions. But the collection of open sets in the topology determined by Definition 1-1 has only cardinal number c. A proof of this is presented later.

    It is natural to ask if we can select a still smaller collection of subsets and use these to define the open sets. The answer is often affirmative, and the following definition provides such a collection.

    A collection of subsets {Bα} of a given set S is a basis for a topology in S provided that

    Bα = S and that

    (2) if p is a point of Bα Bβ, then there is an element Bδ of {Bα} which contains p and which itself is contained in Bα Bβ.

    We note that the collection of open sets satisfying Axioms O1, O2, and O3 is a basis according to this definition.

    = {Bα} is such a basis in a set S. We define open set, and hence a topology, by agreeing that a subset of S . The resulting collection of open sets satisfies Axiom O3, for by (1) S is open, and by agreement ø is open. Also, the satisfaction of Axiom O1 is obvious, for a union of unions of basis elements is a union of basis elements. To establish Axiom O2, we first point out that condition (2) can be formulated as follows:

    (2′) If p that contains p is any finite collection of basis elements, then for each point p containing p But since Bα(p) contains p for each point ppBα(p) Thus this intersection is a union of basis elements and is open. The same kind of argument will also show that the intersection of a finite number of open sets is a union of basis elements and hence is open. We may therefore state the following result.

    satisfies the axioms for a topological space.

    Given a set S and some intuitive idea of what its topology should be, it is usually much easier to find a basis that agrees with the intuition than it is to describe the open sets in general. However, there may be many choices for a basis, all giving the same topology. For example, in the Euclidean plane we can take as a basis the collection of all interiors of circles or the set of all interiors of squares. Since any union of interiors of circles is a union of interiors of squares and conversely, it is obvious that both collections define the same open sets in the plane. Either of these collections defines the Euclidean topology for the plane. We also could have used as a basis the collection of all interiors of ellipses, or all interiors of triangles, or all interiors of crescents, and achieved the same topology. This is an example of the equivalence of different bases.

    Two bases are equivalent if they determine identical collections of open sets.

    ′ for topologies in a set S be equivalent is that if p is a point of an element B , then there is an element B′ containing the point p and contained in B and conversely.

    Proof: ′ are equivalent, then the condition is obviously satisfied. Suppose that the condition holds, and let O . Then each point of O ′, and this element is contained in O. Thus O . ⁡

    Of course, it is also possible to choose nonequivalent bases, but this will lead to differed topologies. For instance, the set of all half-planes x > x0 for all real numbers x0 satisfies the two conditions for a basis for a topology of the plane. It is easy to see that the only nonempty open sets of this topology are the plane itself and the elements of the basis. It is true that each such open set is open in the Euclidean topology, but the Euclidean topology has many open sets that are not open in this new topology. Thus the two topologies are not equivalent, although in a sense to be discussed shortly, they are comparable.

    Another example of a different basis for the plane is the set of all horizontal open line segments. It is left as an easy exercise to show that every Euclidean open set is open in this new topology but not conversely. That is, there are more open sets in this new topology than in the Euclidean topology.

    It is often the case that we have a topological space S but still find it convenient to select a basis for S. That is, we choose a particular subcollection of the open subsets of S as a basis, in such a way that the new basis is equivalent to the basis of all open sets of S. A subcollection (B of open sets of a topological space S is a basis for S if and only if every open set in S is a union of elements of (B. (This is a slightly different use of the word basis than that given by the previous definition, but we will not discriminate between them.) The concept of a countable basis illustrates this situation. A countable basis for a space S is a basis that contains only countably many sets. This term is used almost always in the sense of a basis for a topology already given in S.

    EXERCISE 1-3. The collection of all circles in the plane with rational radii and with centers having rational coordinates is a countable collection. Show that the interiors of such circles form a basis for the Euclidean topology of the plane.

    Now suppose we have a set S and any collection {Xα} of subsets of S such that UαXα = S.S. Can we define a topology for S in which each Xα is an open set? The answer is yes because we may always assign to S the discrete topology in which there are no limit points. In the discrete topology, every set is closed and hence every set is open. It appears that our question should have been, "Is there a topology for S in which each Xα is open, and in which there are no ‘extraneous′ open sets?" By this we mean that no proper subcollection of the open sets contains all of the sets Xα and satisfies Axioms O1, O2, and O3. The answer is still yes. Any collection of sets satisfying Axiom 02 and containing all the sets Xα must also contain all finite intersections of sets in {Xα}. Then if the same collection satisfies Axiom O. of all finite intersections of sets in {Xα} (each Xα is such an intersection) satisfies the conditions for a basis and hence determines a collection 0 of open sets for a topology in S. The topology so determined answers our question affirmatively, and this situation motivates our next definition.

    . of all open sets of a topological space S is a subbasis of S is a basis for S.

    EXERCISE 1-4. Show that the collection of all open half-planes is a subbasis for the Euclidean topology of the plane.

    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.

    Let {Oα} and {Rα} be two collections of subsets of a set S, both satisfying Axioms O1, 02, and 02,. That is, S has two topologies. We will say that the topology J1 determined by {Oα} is a finer topology than the topology j2 determined by {Rβ} if every set Rβ is a union of sets Oα, that is, each Rβ is open in the j1 topology. We will denote this situation with the symbol jj2. We easily see that the two topologies are equivalent if we have both jj2. We now consider the collection of all possible topologies on a given set S. As an exercise, the reader may prove the following result due to Birkhoff [63]: the collection of all topologies on a given set S constitutes a lattice under the partial ordering defined above.

    1-4 Metric spaces and metric topologies. In this section, we give the most direct generalization of the topology used in real numbers in analysis.

    Let M be a set of points, and assume that there exists a real-valued function d(x, y) on pairs of elements of M satisfying the following conditions:

    1. d(x, y) 0.

    2. d(x, y) = 0 if and only if x = y.

    3. d(x,y) = d(y, x).

    4. d(x, y) + d(y, z) d(x, z) (the triangle inequality).

    We say that M is a metric space with metric d, or with distance function d.

    The spaces that are most familiar to the reader are metric spaces. For example, if we define the distance between two real numbers x and y by setting d(x, y) = |x — y|, we have converted the real numbers into a metric space.

    A metric provides an easy way to define a topology in a metric space. For let x be any point of a metric space M with metric d, and let r be a positive number. The spherical neighborhood S(x, r) of the point x is the set of all points y in M such that d(x, y) r, the number r being the radius of the neighborhood.

    The set of all spherical neighborhoods in M satisfies the conditions for a basis. The first condition is satisfied trivially, of course. To prove that the second condition holds, let p be any point in an intersection S(x1, r1) S(x2, r2). Let r be the smaller of the two numbers r1 — d(p, x1) and r2 — d(p, x2). Since p is in both spherical neighborhoods, it follows that r is positive. Now suppose q is a point in S(p, r). Then for i = 1 or 2,

    we have

    Thus q lies in S(xi, ri), i — 1,2, and hence S(p, r) is contained in the intersection S(x1 rS(x2, r2).

    The topology defined in a metric space M by the basis of all spherical neighborhoods in M is the metric topology of M.

    As an important example we define Euclidean n-dimensional space En. The points of En are all ordered n-tuples (x1 x2, … , xn) of real numbers. If x = ((x1, … , xn)) and y = (yl … , yn), then we define

    It is left as an exercise for the reader to prove that this is indeed a metric. (It is evident that we are using nothing more than the usual formula for the distance between two points as we find it in analytic geometry.)

    One may consider a metric space from two standpoints. To the topologist, the particular metric used on a space is merely a convenient way to define open sets. For instance, we may use the metric

    for En and obtain exactly the Euclidean topology. The metric is often a convenience in proving theorems and, to the topologist, the choice between equivalent metrics is merely a question of expediency.

    On the other hand, to a metric geometer the metric is important in itself. A change in the metric changes the metric space. As we pointed out earlier, the natural equivalence relation between topological spaces is the homeomorphism. For the geometer, the corresponding transformation on metric spaces is the isometry, a one-to-one distance-preserving transformation of one metric space onto another. A question that might interest a metric geometer is this: does a certain type of metric space M with metric d have the midpoint property, i.e., for each two points x and y in M is there a point z such that d(x, z) = d(y, z) ½d(x, y)? This is not a topological question at all. To see this, we note that the closed interval [0, 1] in E¹ and the closed semicircle p π (in polar coordinates) are homeomorphic under the homeomorphism f(x) = (l,πx). Then in the Euclidean metrics the first example has the midpoint property and the second does not. A topologist might be interested in knowing whether a certain type of space has at least one metric with the midpoint property or if the space were such that every metric has the midpoint property.

    We will use the term metric space to mean a topological space that has a metric such that the basis of spherical neighborhoods yields the original topology. Of course, any set may be assigned a distance function. We simply let the distance between distinct points equal unity in every case, and the axioms for a metric will be satisfied. This metric will impose the discrete topology on the set, however. The crux of the matter here is the requirement that the metric topology be the original topology. In this sense, a metric space is often called a metrizable space.

    As an example of the topological power of a metric, we give the following result. First, a set X of points in a space S is said to be dense in S if every point of S is a point or a limit point of X, that is, if S = X. A space is separable if it has a countable dense subset. For instance, En is separable since the set of all points whose coordinates are all rational is countable and dense.

    THEOREM 1-5. Every separable metric space has a countable basis.

    Proof: Let M be a metric space with metric d(x, y) and having a countable dense subset X = {xi}. For each rational number r > 0 and each integer i > 0, there is a spherical neighborhood S(xiis a basis. Let p be any point of M and let 0 be an open set containing psuch that S(p) is contained in 0, by definition. There is a point xi of X such that d(xi, p) /3 since X is dense. Let r /3, and consider S(xi, r) Certainly S(xi, r) contains p, and if y is any point of S(xi, r), then

    Thus y is in S(p) and so S(xi, r) that contains p and lies in 0. It follows that 0 is a basis for the topology of M

    Without the assumption of metricity, Theorem 1-5 is not true. In E², consider the set P of all points (x, y) with y 0. Let a basis for P consist of (1) all interiors of circles in P but not touching the x-axis, and (2) the union of a point on the x-axis and the interior of a circle tangent from above to the x-axis at that point. The set of points in P for Psuch that each point of the x-axis lies in one and only one element of that subcollection. This contradicts the fact the real numbers are uncountable.

    EXERCISE 1-6. In En, let x = (x1 …, xn) and y = (y1 …, yn), and define d′(x, y) = Σni=1 |xi — yi| and d″(x, y) = maxi |xi — yi|. Show that both df and d" give the same topology as the Euclidean metric. What do the basis elements look like?

    1-5 Continuous mappings. The definition of a continuous transformation given in Section 1-1 is not easy to apply. A more useful criterion for continuity is contained in Theorem 1-6. In fact, this condition is usually given as the definition of continuity.

    EXERCISE 1-6. Let f:S → T be a transformation of the space S into the space T. A necessary and sufficient condition that / be continuous is that if 0 is any open subset of T, then its inverse image F–1(0) is open in S.

    [Note that to speak of F_1(0), it is not necessary that each point of 0 be the image of a point of S. Indeed, F_1(0) may very well be empty.]

    Proof: Suppose first that f is continuous, and let 0 be open in T. If f –¹(0) is not open, then S — f –¹(0) is not closed. Hence there is some point p in f –¹(0) that is a limit point of S — f –¹(0). By the definition of continuity, f(p) is a limit point or a point of f[S — f –¹(0)]. It is certainly possible that disjoint sets have intersecting images in general, but not if one of these is an inverse set. That is, we can assert that f[f –¹(0)] C 0 and F[S — f –¹(0)] are disjoint. This implies that F(p) cannot be a point of F[S — f –¹(0)], so it must be a limit point of this set. But 0 is an open set of T that contains f(p) but no point of F[S — f –¹(0)]. This contradicts the definition of limit point, and hence Ff –¹(0) must be open.

    The argument in the other direction is even easier. Suppose p is a limit point of a subset X of S. If f(p) is not in f(X), then T — f(X) is an open set containing F(p). Hence f –¹[T — F(X)] is an open set containing p but not intersecting X

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

    Proof: To establish the sufficiency, we show that if 0 is an open set in T, then f –¹(0) is open in S. To do so, let x be a point of f –¹(0). Then 0 is an open set containing f(x) so that there is an open set Ux containing x and such that f(Ux) lies in 0. It follows that Ux is in f –¹(0) and that f –¹(0) = X UX. Hence f –¹(0) is open. For the necessity, take U =f –¹(V)

    EXERCISE 1-7. Show that a one-to-one transformation f:S → T of a space S onto a space T is a homeomorphism if and only if both F and f-1 are continuous.

    A rewording of Theorem 1-7 for metric spaces strongly resembles the classic definition of continuity in analysis.

    THEOREM 1-8. Let F:M → N be a transformation of the metric space M with metric d into the metric space N with metric p. A necessary and sufficient condition that F is any positive number and x is a point of M, then there is a number δ > 0 such that if d(x,y) < δ, then p[f(x), f(y)] < .

    Proof: The sufficiency of the condition is easily established. Let V be an open set in N and y be a point of V. There is a spherical neighborhood S(y, ) lying in V. The given condition implies that the neighborhood S(x, δ), x in f –¹(y), in M is such that f[S(x, δ)] is contained in S(y, ) and hence lies in V. Thus the condition of

    At an early stage in his study of topology, the student may not recall whether Theorem 1-6 says that the inverse of an open set is open or that the image of an open set is open. Both conditions seem equally sensible. It may help to give a name to the second possibility, which is, moreover, an important type of transformation.

    A transformation f:S → 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 subset of S is open in T.

    Some writers discuss transformations that carry open sets into open sets but that are not necessarily continuous. Such transformations are usually called open.

    We will refer to a continuous transformation as a mapping from now on.

    THEOREM 1-9. A necessary and sufficient condition that the one-to-one mapping f:S → T of the space S onto the space T be a homeomorphism is that F be interior.

    Proof: According to Exercise 1-7, we need only show that f –¹ is continuous. But this follows from Theorem 1-6, for if 0 is open in S, then (f –¹)_1 (0) = f(0) is open in T. Thus f

    It is now easy to give examples of one-to-one mappings that are not homeomorphisms. Let S be the set of all nonnegative real numbers with their metric topology, and let T be the unit circle in its metric topology. For each x in S, let f(x) = (1, 2πx²/(l + x²)), 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 and is not a homeomorphism.

    1-6 Connectedness. Subspace topologies. Perhaps the reader feels that some examples of useful topological results are overdue. One important example of the usefulness of our development is embodied in this section.

    A topological space is separated if it is the union of two disjoint, nonempty open sets. A space is connected if it is not separated. It should be obvious that either property is invariant under a homeomorphism.

    We may leave the proofs of the following lemmas as exercises:

    LEMMA 1-10. A space is separated if and only if it is the union of two disjoint, nonempty closed sets.

    LEMMA 1-11. A space S is connected if and only if the only sets in S that are both open and closed are S and the empty set.

    THEOREM 1-12. The real line E¹ is connected.

    To prove such a theorem, we must use some properties of the real number system. We have assumed implicitly that the reader already knows a good deal about the real numbers, and we do not intend to make a detailed study here. We do state one important property, however, and take it to be an axiom.

    DEDEKIND CUT AXIOM. Let L and R be two subsets of E¹ with the three properties that (1) neither L nor R is empty, (2) R U L = E¹, and (3) every number in L is less than any number of R. Then there is either a largest number in L or else a smallest number in R, but not both.

    Proof of Theorem 1-12. Suppose E¹ is not connected. Then it is the union of two disjoint nonempty open sets, U and V. Let u be some point in U and v be some point in V. It is, at most, a renaming of the sets to assume that u < v. Let L consist of (a) all numbers, whether in U or V, that are less than u} together with (b) all numbers x such that every point in the closed interval [u, x] belongs to U. Let R be all other numbers. Certainly L is nonempty, and since v must lie in R, R is also nonempty. By definition, every number is in L or in R. Also, by construction, every number in L is less than every number in R. Thus L and R form a Dedekind cut, and there is a number m that is either the largest in L or the smallest in R. The number m must lie in U or in V; suppose first that m is in U. Then there is an open interval (a, b) containing m and lying in the open set U. We may assume that a and b are also in U. If m is in L, then m u, and we have u m < b. But then [u, b] lies in U, so b is in L, although m < b. Hence m cannot be in L. If m is the smallest number in R, there must be a point y of V between u and m. But then y must be in R, although y is less than m, another contradiction. Thus m cannot be in U. If m belongs to V, we can choose [a, b] to lie in V with a < m < b. If m is in R} then a is in J and is less than m. If m lies in L, then b is also in L and is greater than m. Hence m cannot be in V. This means we have a contradiction in any case, so E¹

    We have defined a connected space, but it should be obvious that there are separated spaces that contain connected sets. For instance, consider the union of two parallel lines. There is a general principle for changing a definition so that it applies to a subset of a space.

    Let S be a topological space and X be a subset of S. The subspace topology of X is that obtained by defining a subset U of X to be open in X if it is the intersection of X with some open subset of S. That is, we take for open sets of X all sets of the form X 0, where 0 is open in S. It is easy to prove that, with this topology, X is a topological space, a subspace of S. This implies that we have here a general method for constructing many topological spaces.

    Furthermore, we can now say that a property defined for spaces is a property of a subset X if X has the property as a subspace. Thus X is a connected subset of a space as if X is a connected subspace of S. Expressed without using subspace topology, this says that a subset X of S is connected if there do not exist two open sets U and V in S such that U X and V X are disjoint and nonempty, and such that (U X) (V X) = X.

    The subspace topology is also called the relative topology. We speak of a subset A of a subset X of a space S as being open relative

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