TOPOLOGY 541Let M be a set with two members a and b. Define the fu.pdf
- 1. TOPOLOGY 541 Let M be a set with two members a and b. Define the function D : M × M as follows: D(a,a) = D(b,b) = 0,D(a,b) = D(b,a) = r where r is a positive real number. Prove that (M, D) is a metric space. Solution Definition 1.1 (Metric). A metric, or distance function, on a set X is a mapping d : X × X R such that • d(x, y) 0 for all x, y X, and d(x, y) = 0 if and only if x = y. • d(x, y) = d(y, x) for all x, y X. • d(x, z) d(x, y) + d(y, z) for all x, y, z X. We call (X, d) a metric space. Definition 1.2 (Open ball). Let (X, d) be a metric space. For x X and > 0, the set Bd(x, ) defined by Bd(x, ) = {y X | d(x, y) < } is callen an open ball in the set X. Definition 1.3 (Open sets in metric spaces). Let (X, d) be a metric space and let U be any subset of X. Then U is called an open set in X if every point of U is an interior point of U; that is, for any a U, there is an open ball B(a, ) such that B(a, ) U. Definition 1.4 (Properties of open sets). Let (X, d) be a metric space. • and X are open. • The union of an arbitrary collection of open sets is open. • The intersection of a finite number of open sets is open. Definition 1.5 (Closed set). A subset A of a metric space (X, d) is closed if it’s complement XA is open in X. Definition 1.6 (Properties of closed sets). Let (X, d) be a metric space. • and X are closed. • The union of an finite collection of closed sets is closed. • The intersection of an arbitrary number of closed sets is closed. Definition 1.7 (Limit point of a subset). Let (X, d) be a metric space and let A be a subset of X. Then a point x in X is a limit point of A if every open ball B(x, ) contains at least one point of A. The set of all limit points of A is called the derived set A . Definition 1.8 (Closure of a set). Let (X, d) be a metric space and let A X. Then the set consisting of A and its limit points is called the closure of A, denoted A. A = A A Corollary 1.19. We note that A is dense in X if and only if for any x X and > 0, there is a point a A such that d(x, a) < . 1.2. Subspaces. Definition 1.20 (Open sets in a subspace). Let (X, d) be a metric space and (Y, dY ) be a metric subspace of (X, d). Let G be a subset of Y . Then G is open in Y if and only if, for any x G, there is an open ball B(x, ) in X such that B(x, ) Y G A subset H of Y is closed in Y if its complement G = Y H of H is open in Y . Theorem 1.21 (Open sets in a metric subspace). Let (Y, dY ) be a metric subspace of a metric space (X, d), and let G Y . Then G is open in Y if and only if there exists an open subset U in X such that G = U Y . 1.3. Convergence in a Metric Space. Definition 1.22 (Convergence). A sequence (xn) in a metric space (X, d) is said to converge to a point x X if for any > 0, there exists N such that n > N implies d(xn, x) < The point x is called a limit of the sequence (xn) Corollary 1.23. A
- 2. sequence (xn) in a metric space (X, d) is said to converge to a point x X if any open ball B(x, ) contains almost all xn. Theorem 1.24 (Connection between closed sets and convergent sequences). Let (X, d) be a metric space, A X and x X. Then • x A if and only if there is a sequence (xn) in A such that xn x. • A is closed if and only if A contains all the limits of convergent sequences in A. Definition 1.25 (Uniform convergence). Let (fn) be a sequence of real-valued functions defined on a set S and let f be a function defined on S. Then we say that the sequence (fn) converges to f uniformly if for any > 0, there exists N such that sup xS d(fn(x), f(x)) < for all n > N, and where N is independent of x. Definition 1.26 (Cauchy Sequences). A sequence (xn) in a metric space (X, d) is said to be Cauchy in X if for any > 0, there exists N such that m, n > N d(xm, xn) < If a homeomorphism exists between X and Y , we say that X and Y are homeomorphic, and that X Y . Definition 3.2 (Characterisations of homeomorphism). Let f : X Y be a bijective mapping. Then the following are equivalent. • f is a homeomorphism; • for any U X, U is open in X if and only if f(U) is open in Y ; • for any G X, G is closed in X if and only if f(G) is closed in Y ; • for any A X, f(A) = f(A); • for any B Y , f 1(B) = f 1 (B) • for any B Y , f 1 (Int B) = Int f 1 (B) Definition 3.3 (Isometric mappings). Let (X, d) and (Y, dY ) be two metric spaces and f : X Y a mapping. Then f is said to be isometric or an isometry if f preserves distances; that is, for all x, y X, dY (f(x), f(y)) = dX(x, y) The space X is said to be isometric with the space Y if there exists a bijective isometry of X onto Y . The spaces X and Y are isometric spaces Theorem 3.4. Any isometric mapping from X onto Y is a homeomorphism. Moreover, if X is complete and Y is isometric with X, then Y is also complete. Definition 3.5 (Equivalent metrics). Let (X, d1) and (X, d2) be two metric spaces. If the identity mapping id : (X, d1) (X, d2) is a homeomorphism, then the metrics d1 and d2 are said to be equivalent on X. Theorem 3.6 (Characterisations of equivalent metrics). Let (X, d1) and (X, d2) be two metric spaces. Then the following are equivalent. • The metrics d1 and d2 are equivalent on X; • for any U X, U is open in (X, d1) if and only if U is open in (X, d2); • for any G X, G is closed in (X, d1) if and only if G is closed in (X, d2); • The sequence (xn) converges to a in (X, d1) if and only if it converges to a in (X, d2). Theorem 3.7 (Equivalent metrics). Let (X, d1) and (X, d2) be two metric spaces. If there exist strictly positive numbers c and C such that cd1(x, y) d2(x, y) Cd1(x, y) for all x, y X, then the metrics d1, d2 are equivalent on X.