User:J D Bowen/Math710 HW2

==working on now 1) We aim to show that a function $$f:E\subset(X,\sigma)\to(Y,\rho) \ $$ is continuous if and only if $$\lim_{n\to\infty} f(x_n)=f(x) \ $$ whenever $$\left\{{x_n}\right\} \subset E \ $$ and $$x_n \to x \ $$.

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2) Let $$E\subset X \ $$ be a compact subset of a metric space and let $$f:E\to\mathbb{R} \ $$ be continuous. We aim to show $$f \ $$ is uniformly continuous.

3) Let $$(X,d) \ $$ be a metric space and $$G\subseteq E \subseteq X \ $$. We say $$G \ $$ is relatively open in $$E \ $$ if $$G=O\cap E \ $$ for some open subset $$O \ $$ of $$X \ $$.  We aim to show that $$G \ $$ is relatively open in $$E \ $$ if and only if $$\forall x \in G, \exists \delta>0 :B_\delta(x)\cap E \subseteq G \ $$.

4) Let $$(X, d), (Y, \rho) \ $$ be metric spaces. Suppose $$E \subseteq X, \ f : E \to Y \ $$. Prove that f is continuous if $$f^{-1} (U)$$ is relatively open in E whenever U is open in Y.

6) Let $$(X, d) \ $$ be a metric space with $$\varnothing \neq A \subseteq X \ $$. Define $$f : X \to R \ $$ by

$$f(x) = \text{inf}_{a\in A} d(x,a) \ $$.

(a) What is the zero set of f ? (b) Prove that f is continuous on X

7) Let $$(X, d) \ $$ be an incomplete metric space. Let $$\hat{X} \ $$ denote the set of all Cauchy sequences in $$X \ $$. Define a relation $$~ \ $$ on $$\hat{X} \ $$ by $$\left\{{x_n}\right\} ~ \left\{{y_n}\right\} \iff \lim_{n\to\infty} d(x_n,y_n)=0 \ $$.

(a) Show that this an equivalence relation. (b) Let $$X^* \ $$ denote the equivalence classes on $$\hat{X} \ $$ defined by $$~ \ $$. For $$[\left\{{x_n}\right\}], [\left\{{y_n}\right\}] \in X^* \ $$, define

$$d^* ([\left\{{x_n}\right\}], [\left\{{y_n}\right\}]) = \lim_{n\to\infty} d(x_n,y_n) \ $$.

Show $$(X^*, d^*) \ $$ is a complete metric space. (c) Define the map $$j:X\to X^* \ $$ by $$j(x)=[\left\{{x_n}\right\}] \ $$ where $$x_n=x \ \forall n \ $$. Show that $$d^*(j(x),j(y)) = d(x,y) \ $$ and $$\overline{j(X)}=X^* \ $$.