Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
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Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X...
Let (X, d) be a discrete space and let (Y, d′) be any metric space. Prove that any function f : (X, d) → (Y, d′) is continuous. (Namely, any function from a discrete space to any metric space is continuous.)
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.
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
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) 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...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Let (Q, d) be the metric space consisting of the set Q of rational numbers with the standard metric d(x, y) = |x-yl. Show that the Heine Borel theorem fails for (Q,d). In other words, show that (Q, d) has a subset SCQ that is closed and bounded, but not compact
Let (Q, d) be the metric space consisting of the set Q of rational numbers with the standard metric d(x, y) = |x-yl. Show that the Heine Borel theorem fails for (Q,d). In other words, show that (Q, d) has a subset SCQ that is closed and bounded, but not compact
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...
1. Let (Q, d) be the metric space consisting of the set Q of rational numbers with the standard metric d(x, y) = (x – yl. Show that the Heine-Borel theorem fails for (Q, d). In other words, show that (Q, d) has a subset SCQ that is closed and bounded, but not compact (8 points).
Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements (1) If a sequence is convergent in X, all its subsequences are converging to the same limit as the original sequence. (2) If a subsequence of a Cauchy sequence is convergent, then the whole sequence is convergent to the same limit as the subsequence. (3) Suppose that (X,d) is complete and Y S X is closed in (X,d). Then the space (Y,dlyxy) is complete....