5. Let (S,d) be a metric space. Let E c S. Prove or disprove: if E-d(E),...
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.)
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
(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
(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.
5. Let X be a metric space. (a) Let x E X be an isolated point. Prove that the only sequences in X that converge to a are the sequences that are eventually constant (b) Prove that the only convergent sequences in a discrete metric space (See Problem 8 on page 79 for the definitions of "isolated" and "discrete.") with tail a,z,x.... are the eventually constant sequences.
Let (X, d) be a compact metric space. Prove that if F ⊆ C(X) is equicontinuous then it is uniformly equicontinuous.
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....
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...
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...
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...