Let (X, d) be a metric space, and let ACX be a subset (a) (3 pts) Let x E X. Write the definition of d(x, A) (b) (7 pts) Assume A is closed. Prove that d(x,A-0 if and only if x E A. Let (X, d) b...
(1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated (1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
Adams-Franzosa 5.25) Let (X, dx) be a metric space. Assume pE X and AcX (a) Find an example showing that d(p). A) = 0 need not imply that p Prove that you indeed have an example (b) Prove that if A is closed and d(p), A) 0, then pE A (c) Prove that for any p and A, if pE A, then d(p), A) 0 Adams-Franzosa 5.25) Let (X, dx) be a metric space. Assume pE X and AcX (a)...
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Closed sets. A subset S of a metric space M is closed, if its complement S is open. A closed ball in a metric space M, with center xo and radius r> 0, is the set of points В, (хо) %3D {x € M: d (x, хо) < r}. Problem 6.4. Prove that, for any metric space E, the entire space E is a closed set.
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if 3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
6 6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If T1 is the topology induced orn Y by di and T2 is the subspace topology on Y (induced by T on X), prove that Ti -T2. [This shows that every subspace...
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is a compact subset of X, and Z C Y, then Z is compact if and only if Z is closed (b) IfY. Y are a finite collection of compact subsets of X, then their union Y1 U...UYn is also compact. (c) Every finite subset of X (including the empty set) is compact.
Prove the Theorem: Let A and B be regularly closed sets in a metric space X. If aAnBº + then Aºn B° + Ø.
Let X be a metric space and let E C X. The boundary aE of E is defined by E EnE (a) Prove that DE = E\ E°. Here Eo is the set of all interior points of E; E° is called the interior of E (b) Prove that E is open if and only if EnaE Ø. (c) Prove that E is closed if and only if aE C E (d) For X R find Q (e) For X...