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CHAPTER 1
Topological Spaces and Groups
1.0. Introduction
This chapter contains the preliminary and somewhat elementary facts of general spaces and groups. Proofs are given in considerable detail and there are examples which may be of help to a reader for whom the subject is new.
We use the standard set-theoretic symbols: capitals A, B, etc. for sets, A [union] B for the union of sets (elements in one or both), A [intersection] B for the intersection (elements in both), etc.
1.1. Spaces
The term space is sometimes used in mathematical literature in a very general sense to denote any collection whose individual objects are called points, but in topology the term space is used only when some further structure is specified for the collection. As the term will be used in this book it has a meaning which is convenient in studying topological groups. The definition is as follows:
Definition. A topological space (or more simply space) is a non-empty set of points certain subsets of which are designated as open and where, moreover, these open sets are subject to the following conditions:
1) The intersection of any finite number of open sets is open.
2) The union of any number of open sets is open.
3) The empty set and the whole space are open.
4) To each pair of distinct points of space there is associated at least one open set which contains one of the points and does not contain the other.
A space is called discrete if each point is an open set.
Condition 4) is known as the T0-separation axiom in the terminology of Alexandroff and Hopf. The first three conditions define a topological space in their terminology. The designated system of open sets is the essential part of the topology, and the same set of points can become a topological space in many ways by choosing different systems of subsets designated as open.
1.2. Homeomorphisms
Definition. A homeomorphism is a one-one relation between all points of one topological space and all points of a second which puts the open sets of the two spaces in one-one correspondence; the spaces are topologically equivalent.
The notion of homeomorphism is reflexive, symmetric, and transitive so that it is an equivalence relation in a given set of topological spaces.
Examples of spaces. Let E1 denote the set of all real numbers in its customary topology: the open intervals are the sets {y; x < y < z} for every x z. The open sets are those which are unions of open intervals together with the empty-set (null-set) and the whole space.
Let R1 [subset] E1 denote the set of numbers in the closed interval 0 ≤ y ≤ 1, where for the moment we take this subset without a topology. This set gives distinct spaces as follows:
1) Topologize R1 as customarily: the open sets are the intersections of R1 with the open sets of E1.
2) Topologize R1 discretely, that is let every subset be open.
3) Topologize R1 by the choice: open sets are the null set, the whole space and for each x of R1 the set {z, x z ≤ 1}.
4) Topologize R1 by the choice: the complement of any finite set is open and the empty set and the whole space are open.
In the sequel R1 will denote the closed unit interval and E1 the set of all reals in the customary topology. A set homeomorphic to R1 is called an arc. A set homeomorphic to a circle is called a simple closed curve.
1.3. Basis
Definition. A collection {Qa} of open sets of a space is called a basis for open sets if every open set (except possibly the null set) in the space can be represented as a union of sets in {Q0}. It is called a sub-basis if every open set can be represented as a union of finite intersections of sets in {Qa} (except possibly the null set).
A collection {Qa} of open sets of a space S is a basis if and only if for every open set Q in S and x [member of] Q there is a Qa [member of] {Qa} such that
x [member of] Qa [subset] Q.
If a collection has this property at a particular point x then the collection is called a basis at x.
If a set together with certain subsets are called a sub-basis, then another family of subsets is determined from the sub-basis by taking arbitrary unions and finite intersections. This new family (with the null set added if necessary) then satisfies conditions 1), 2), 3) for the open sets of a topological space. Whether 4) will also be satisfied depends on the original family of sets.
Example. Let E1 denote the space of real numbers in its usual topology. For each pair of rationals r1< r2 let (r1, r2) denote the set of reals r1< x r2. This countable collection of open sets is a basis.
A space is said to be separable or to satisfy the second countability axiom if it has a countable basis. A space is said to satisfy the "first countability axiom" if it has a countable basis at each point.
Example. Let S denote a topological space and let F denote a collection of real valued functions f(x), x [member of] S. If f0 is a particular element of F then for each positive integer n let Q(f0, n) = {f [member of] F; [absolute value of (f(x) - f0(x))] < 1/n. We may topologize F by choosing the sets Q(f0, n) for all f0 and n as a sub-basis. The topological space so obtained has a countable basis at each point in many important cases.
1.4. Topology of subsets
Let S be a topological space, T a subset. Let Q [intersection] T be called open in T or open relative to T if Q is open in S. With open sets defined in this way T becomes a topological space and the topology so defined in T is called the induced or relative topology. If S has a countable basis and T [subset] S then T has a countable basis in the induced topology.
Definition. A subset X [subset] S is called closed if the complement S - X is open. If X [subset] T [subset] S, X is called closed in T when T - X is open in T.
Notice that T closed in S and X closed in T implies that X is closed in S. The corresponding assertion for relatively open sets is also true.
It can be seen that finite unions and arbitrary intersections of closed sets are again closed.
Definition. If K [subset] S, the intersection of all closed subsets of S which contain K is called the closure of K and is denoted by [bar.K]. If K is closed, K = [bar.K].
1.5. Continuous maps
Let S and T be spaces (= topological spaces) and f a map of S into T
f : s [right arrow] T
that is, for each x in S, y = f(x) is a point of T. If the inverse of each relatively open set in f(S) is an open set in S then f is called continuous. In case f(S) = T then f continuous and V open in T imply f-1(V) is open in S. The map is called an open map if it carries open sets to open sets.
If f is a continuous map of S onto T (that is f(S) = T) and if f-1 is also single valued and continuous, then f and f-1 are homeomorphisms and S and T are homeomorphic or topologically equivalent (1.2).
Example. The map f(t) = exp (2π[square root of (-1 t)]) is a continuous and open map of E1 onto a circle (circumference) in the complex plane.
Example. Let K denote the cylindrical surface, described in x, y, z coordinates in three-space by x2 + y2 = 1. Let f1 denote the map of K onto E1 given by (x, y, z) [right arrow] (0, 0, z), let f2 denote the map of K that is given by (x, y, z) [right arrow] (x, y, 0) and f3 the map (x, y, z) [right arrow] (x, y, [absolute value of (z)]) ot K into K. All three maps are continuous, the first two are open, and f1 and f3 are also closed, i.e. they map closed sets into closed sets.
1.6. Topological products
The space of n real variables (x1, x2, ..., xn), - ∞ < xi ∞, i = 1, ..., n, and the cylinder K of the preceding example are instances of topological products.
Let A denote any non-null set of indices and suppose that to each a [member of] A there is associated a topological space Sa. The totality of functions f defined on A such that f(a) [member of] Sa, for each a [member of] A, is called the product of the spaces Sa. When topologized as below it will be denoted by PROD Sa; we also use the standard symbol ×, thus E × B is the set of ordered pairs (e, b), e [member of] E, b [member of] B.
The standard topology for this product space is defined as follows. For each positive integer n, for each choice of n indices a1, a2, ..., an, and for each choice of a non-empty open set in Sai.
[MATHEMATICAL EXPRESSION OMITTED]
consider the set of functions f [member of] PROD Sa for which
[MATHEMATICAL EXPRESSION OMITTED]
Let the totality of these sets be a sub-basis for the product. The resulting family of open sets satisfies the definition of space in 1.1.
Example 1. The space En = E1 × E1 × ... × E1, n copies, is the space of n real variables; here A = {1, 2, ..., n} and each Si is homeomorphic to E1 (1.2). Let xi [member of] Si. Then (x1, ..., xn) are the "coordinates" of a point of En. It can be verified that the sets Um(x), m [member of] I (the collection of positive integers), of points of En whose euclidean distance from x = (x1, ..., xn) is less than 1/m, form a basis at x. The subset R1 × R1 × ... × R1 is an n-cell.
Example 2. Let A be of arbitrary cardinal power and let each Sa, a [member of] A, be homeomorphic to C1, the circumference of a circle. Then PROD Sa is a generalized torus. If A consists of n objects, the product-space is the n-dimensional torus. For n = 2, we get the torus.
Example 3. Let D = S1 × Sa × ... × Sa × ..., n [member of] I, where each is a pair of points – conveniently regarded as the "same" pair, and designated 0 and 2. This is the Cantor Discontinuum, or Cantor Middle Third Set. It is homeomorphic to the subset of the unit interval defined by the convergent series: D : {Σan/3n}, an = 0 or 2. This example will be described in another way in the next section.
Theorem: Let Fa be a closed subset of the topological space Sa, a [member of] A. Then PROD Fa is a closed subset of PROD Sa.
The proof is left to the reader.
1.7. Compactness
Definition. A topological space S is compact if every collection of open sets whose union covers S contains a finite subcollection whose union covers S.
Example 1. The unit interval R1 is compact. Thus let {Q} denote a collection of open sets covering R1. Let F denote the set of points x [member of] R such that the interval 0 ≤ y ≤ x can be covered by a finite subcollection of {Q}. Then F is not empty and is both open and closed. Hence by the Dedekind cut postulate, or the existence of least upper bounds, or the connectedness of R1, it follows that F = R1. To illustrate the concept of compactness consider the open sets Wn [subset] R1, Wn: 1/3n x 1/n, n [member of] I. This collection does not cover R1. Let Wa be the union of two sets: 0 ≤ x < a and 1 - a < x ≤ 1, for some a, 0 < a 1. Now, no matter how a > 0 is chosen, there is always some finite number of the Wn which together with Wa covers R1. Of course R1 minus endpoints is not compact and no finite subcollection of the Wn in this example will cover it.
Theorem. Let S be a compact space and let f: S [right arrow] T be a continuous map of S onto a topological space T. Then T is compact.
Let {Oa} be a covering of T by open sets. Since f is continuous, each f-1(Oa) is an open set in S. There is a finite covering of S by sets of the collection {f-1(Oa)}, and this gives a corresponding finite covering of T by sets of {Oa}. This completes the proof.
Corollary. If f is a continuous map of S into T then f(S)is a compact subset of T.
1.7.1. Theorem. Let S be a compact space and {Da} a collection of closed subsets such that [intersection]aDa is empty. Then there is some finite set [MATHEMATICAL EXPRESSION OMITTED] ..., [MATHEMATICAL EXPRESSION OMITTED] such that [MATHEMATICAL EXPRESSION OMITTED] is empty.
The complement of [intersection]a Da is [union]a(S-Da); if the intersection-set is empty, the union covers S. There is a finite set of indices ai such that S [subset] [union]i([MATHEMATICAL EXPRESSION OMITTED]) and consequently [MATHEMATICAL EXPRESSION OMITTED] is empty, for the same finite set of indices.
Corollary 1. Let Dn, n [member of] I, be a sequence of non-empty closed subsets of the compact space S, with Dn+1 [subset] Dn. Then [intersection]n Dn is not empty.
Application: The Cantor Middle Third Set D. From R1, "delete" the middle third: 1/3 < x 2/3. Let D1 denote the residue: it is a union of two closed intervals. Let D2 denote the closed set in D1 complementary to the union of the middle third intervals: 1/9 < x 2/9 and 7/9 < x 8/9. Continuing inductively, define Dn [subset] Dn-1 consisting of 2n closed mutually exclusive intervals. Let D = [intersection] Dn. This is homeomorphic to the space of Example 3 of 1.6.
Corollary 2. A lower semi-continuous (upper semi-continuous) real-valued function on a compact space has finite g. l. b., greatest lower bound (and l. u. b., least upper bound), and always attains these bounds at some points of space.
This follows from the preceding corollary and the fact that the set where f(x) ≤ r is closed, for every r (similarly, f(x) ≥ r).
1.7.2. Theorem. A topological space with the property: "every collection of closed subsets with empty set-intersection has a finite subcollection whose setinter section is empty", is compact.
The proof, like that of the Theorem of 1.7.1, is based on the duality between open and closed sets.
(Continues…)
Excerpted from "Topological Transformation Groups"
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