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CHAPTER 1
Fundamentals of Topology and Group Theory
In this chapter we collect the relevant definitions and results from topology and group theory to make this book self-contained and easy to read. The material is, indeed, standard and can be found in Bourbaki, Kelley, and Van der Waerden. We shall assume that the reader is familiar with common terms used in set theory (e.g., see Abian).
1. TOPOLOGICAL SPACES
A set X with a family u of its subsets is called a topological space if the following conditions are satisfied: (a) X and [empty set] (null set) are in u; (b) the intersection of any finite number of members of u is in u; (c) the arbitrary union of members of u is in u.
The members of u are called u-open sets of X (or simply open sets of X, if there is no topology other than u in question). A topological space X with a topology u will be denoted by Xu.
For any given set X, there are always two topologies on X. These are: (i) u consisting of all subsets of X. It is easy to check that (a)–(c) are satisfied. This topology is called the discrete topology on X and denoted by d. Xd is called the discrete space.
(ii) u consisting of only X and [empty set]. This topology i is called indiscrete, and Xi is called the indiscrete space.
Let u and v be two topologies on a set X. u is said to be finer than v or, in symbols, u [contains] v if every v-open set is u-open. If u [contains] v then v is said to be coarser than u, or equivalently in symbols, v [contains] u. Furthermore, u [contains] v and v [contains] u if, and only if, u = v (i.e., u is equal to v). Clearly, d is the finest and i the coarsest topology on any set. Any other topology on a set is finer than i and coarser than d.
Let Xu be a topological space and A any subset of X. The largest open set contained in A is called the interior A0 of A. Clearly, a subset A of X is open if, and only if, A = A0.
The complement X ~ U of an open set U in a topological space X is said to be a u-closed or simply a closed set. Using the well-known De Morgan Laws in set theory, one verifies the following: (a') X and [empty set] are closed; (b') the arbitrary intersection of closed sets is closed; (c') a finite union of closed sets is closed.
Let A be a subset of a topological space. The smallest closed set containing A is called the closure [bar.A] of A. To emphasize the topology in which the closure is taken, [bar.A] will be denoted by CluA. The following statements about the closure are immediate: For each subset A of Xu, [MATHEMATICAL EXPRESSION OMITTED] for any finite number of subsets [MATHEMATICAL EXPRESSION OMITTED], and [MATHEMATICAL EXPRESSION OMITTED]. A subset A of Xu is closed if and only if [bar.A] = A.
Let A and B be two subsets of a topological space Xu. A is said to be dense in B if [bar. A] [contains] B. A topological space Xu is said to be separable if Xu contains a countable dense subset.
Let A be any subset of a topological space Xu. Then A can also be topologized as follows: For each u-open set U in Xu, define U [intersection] A to be open in A. Then it is easy to check that the family {U [intersection] A}, when U runs over u, defines a topology on A. This topology on A is called the induced or relative topology of A.
2. METRIC SPACES
An important subclass of topological spaces is the class of metric spaces.
Let E be a set. Suppose there exists a real valued function d defined on the ordered pairs (x, y), x, y [member of] E, satisfying the following axioms:
(m1) d(x, y) ≥ 0, for all x, y [member of] E
(m2) d(x, y) = 0 if and only if x = y
(m3) d(x, y) = d(y, x)
(m4) d(x, y) + d(y, z) ≥ d(x, z) x, y, z [member of] E.
Then E is said to be a metric space. For each positive r > 0 and a fixed x0 [member of] E, the subset Br(x0) = {x [member of] E:d(x, x0) < r} is called on open ball of radius r. The subset {x [member of] E:d(x, x0) ≤ r} is called a closed ball. Let u = {U} be the family of subsets of a metric space E such that, for each U, if x [member of] U then there exists r > 0 such that Br(x) [subset] U. Then it is easy to check that the family u defines a topology on E and E is called a metric topological space or simply a metric space. It is immediate that each open ball of a metric space is an open set and similarly each closed ball is a closed set.
Examples. (1) Let E = R, the real line and d(x, y) = [absolute value of (x - y)]. Then R is a metric space and so a topological space. Apart from this metric topology, indeed, there are other topologies on R as well, e.g., the discrete and the indiscrete.
(2) E = Rn, the space of n-tuples; i.e., x = (x1, ..., xn) where xi [member of] R, 1 ≤ i ≤ n. The metric d is defined by the following formula:
[MATHEMATICAL EXPRESSION OMITTED]
x = (x1, ..., xn) and y = (y1, ..., yn). Then E is a metric space. It is called the Euclidean n-dimensional space. If n = 1, then (2) and (1) coincide.
(3) Let I denote the closed unit interval [0, 1]. Let E = C(I) denote the set of all continuous real-valued functions on I. Define d as follows:
[MATHEMATICAL EXPRESSION OMITTED]
in which f and g are continuous functions on I. Then E is a metric space.
If the topology of a given topological space X can be described by a metric then X is said to be metrizable.
Indeed, as shown above, every arbitrary set can be topologized by at least two topologies, viz., the discrete and the indiscrete. The question of whether a topological space is metrizable is not that easy. Of course, an indiscrete space is not metrizable, as will appear easily from the separation axioms dealt with in the sequel. But a discrete space is metrizable and the metric is the following: For each x, y [member of] X, define
[MATHEMATICAL EXPRESSION OMITTED]
Then it is easy to check that d defines a metric that describes the discrete topology. Sometimes this metric is called a trivial metric.
A sequence {xn} in a metric space is said to be a Cauchy sequence if for each ε > 0 there exists a positive integer n0 depending upon ε such that d(xn, xm) < ε for all n, m ≥ n0. A sequence {xn} in E is said to converge to a point x0 [member of] E if for each ε > 0 there exist n0 = n0(ε) such that d(xn, x0) < ε for all n ≥ n0.
It is easy to show that if a sequence converges then it is a Cauchy sequence. But the converse is not true in general. If each Cauchy sequence converges in a metric space E, then E is said to be a complete metric space. The reader can verify that the metric spaces mentioned in examples (1) to (3) above are all complete metric spaces.
If a metric space E is not complete, then by a well-known procedure it can be completed to a complete space Ê so that E forms a dense subset of Ê. Ê is called the completion of E and is nothing more than the set of all equivalence classes of Cauchy sequences in E. A metric space E is complete if, and only if, it coincides with its completion Ê. (Observe that we do not distinguish between the two sets if their elements can be put in a 1:1 correspondence.)
Observe that the notion of completion depends upon that of Cauchy sequences. Since the latter notion is not necessarily defined on a nonmetric topological space, the notion of completion need not be defined for a nonmetric topological space either. We shall see in the sequel that there is a generalization (uniform spaces) of metric spaces on which the completion can be defined.
In connection with metric spaces the following notions are useful:
A subset A of a metric space E is said to be nowhere-dense or nondense if ([bar.A])0 = [empty set] (i.e., if the interior of the closure of A is empty). A countable union of nondense sets is said to be of the first category. A set that is not of the first category is of the second category. A topological space of the second category is also known as a Baire space.
Theorem 1.(Baire.) Every complete metric space E is of the second category, or is a Baire space.
Proof. Suppose [MATHEMATICAL EXPRESSION OMITTED], where each An is nondense. For each [MATHEMATICAL EXPRESSION OMITTED] is open and everywhere-dense because otherwise there would be an open ball consisting of only points of [bar.An], thus contradicting the fact that the interior of [bar.An] is empty. Since each An is nondense, there exists a sequence {εn} of positive real numbers and a sequence {xn} of elements in E such that [MATHEMATICAL EXPRESSION OMITTED] for each n. We may assume (by induction) that εn+1< εn, εn -> 0 ({εn} converges to zero) and [MATHEMATICAL EXPRESSION OMITTED] for each n ≥ 1. It is easy to see that by this choice {xn} is a Cauchy sequence and, hence, converges to some x0 [member of] E because the latter is complete by hypothesis. Since xm [member of] Bn (xn) [subset] [bar.Bn] (xn) for all m ≥ n, for a fixed [MATHEMATICAL EXPRESSION OMITTED] for each n. Since [bar.Bn] (xn) [intersection] An = [empty set] for each n, x0 [not member of] An for each n ≥ 1, i.e., [MATHEMATICAL EXPRESSION OMITTED], which is a contradiction. Hence, E is of the second category.
It is easy to see from the above theorem that every nonempty open subset of a complete metric space is of the second category.
A subset A in a topological space E is said to be residual if E ~ A is of the first category.
From Theorem 1 it follows that all residual subsets in a complete metric space are nonempty.
3. NEIGHBORHOOD SYSTEMS
Let Xu be a topological space. Let x [member of] X. A subset P of X is said to be a u-neighborhood of x if there exists a u-open set U such that x [member of] U [subset] P. Observe that a neighborhood of a point x [member of] X is not necessarily an open set. However, it is quite clear that an open set is a neighborhood of each point contained in it. The following connection between the open sets and neighborhoods is easy to verify: A subset A of X is u-open if, and only if, for each x [member of] A there exists a neighborhood Px of x such that Px [subset] A.
For each x [member of] Xu, let Ux denote the totality of all u-neighborhoods of x. Then the following properties are immediately established by using the definitions of neighborhoods and open sets:
(n1) For each member Ux in Ux, x [member of] Ux.
(n2) If Ux is in Ux and W is any subset of X such that Ux [subset] W, then W is in Ux.
(n3) Each finite intersection of sets in Ux is also in Ux.
(n4) If Ux is in Ux, then there exists a Vx in Ux such that Vx [subset] Ux and Ux [member of] Uv for each y [member of] Vx, where Uv is the totality of all u-neighborhoods of y. We prove the following:
Proposition 1.Let X be a set. Suppose for each x [member of] X, there exists a system Ux of subsets of X satisfying the above conditions (n1)–(n4). Then there exists a unique topology u on X such that Ux is precisely the system of all uneighborhoods of x for each x [member of] X.
Proof. Let u denote the collection of subsets consisting of [empty set] and of all U such that U [member of] Ux when ever x [member of] U. Then we show that u defines a topology on X. Since for each x [member of] X, X [member of] Ux owing to (n2), u contains X and by definition of u, [empty set] is in u. Let {Ui} (1 ≤ i ≤ n) be a finite family of sets in u. Then for each [MATHEMATICAL EXPRESSION OMITTED], x [member of] Ui for all i, and, hence, Ui [member of] Ux for all i. But then by [MATHEMATICAL EXPRESSION OMITTED]. Finally, let Ua be an arbitrary family of sets in u. Let [MATHEMATICAL EXPRESSION OMITTED]. Then x [member of] Ua for some a and, hence, Ux [member of] Ux. But [MATHEMATICAL EXPRESSION OMITTED] and so [MATHEMATICAL EXPRESSION OMITTED] by (n2). Hence, u defines a topology on X and, for each x, Ux [member of] Ux is a u-neighborhood of x. By using (n4) one sees easily that Ux is precisely the system of all u-neighborhoods of x for each x [member of] X.
(Continues…)
Excerpted from "Introduction to Topological Groups"
by .
Copyright © 1994 Taqdir Husain.
Excerpted by permission of Dover Publications, Inc..
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