General Theory

Let $(X, \rho )$ and $(Y, \sigma )$ be metric spaces. Assume that:

1. $f, f_1, f_2, \ldots$ are bijective functions of $X$ onto $Y$ with inverses $g, g_1, g_2, \ldots$

2. $g$ is uniformly continuous

3. $f_n \to f$ uniformly as $n \to \infty$. Prove that $g_n \to g$ uniformly as $n \to \infty$.

Let $E_1, E_2, \ldots$ be nonempty closed subsets of a complete metric space $(X,d)$ with $E_{n+1} \subset E_n$ for all positive integers $n$, and such that $\lim_{n \to \infty } \operatorname {diam}(E_n) = 0$, where $\operatorname {diam}(E)$ is defined to be

Prove that $\bigcap_{n=1}^{\infty } E_n \neq \emptyset$.

Let $A$ be a subset of a compact metric space $(X,d)$. Assume that, for every continuous function $f : X \to \mathbb {R}$, the restriction of $f$ to $A$ attains a maximum on $A$. Prove that $A$ is compact.

Let $X$ be a metric space and $(x_n)$ a convergent sequence in $X$ with limit $x_0$. Prove that the set $C = \left\{ x_0, x_1, x_2, \ldots \right\}$ is compact.

Prove that every compact metric space has a countable dense subset.

Let $X$ be a compact metric space and $f : X \to X$ an isometry. Show that $f(X) = X$.

Let $M$ be a metric space with metric $d$. Let $C$ be a nonempty closed subset of $M$. Define $f : M \to \mathbb {R}$ by

Show that $f$ is continuous, and that $f(x) = 0$ if and only if $x \in C$.

Let $C^{1/3}$ be the set of real valued functions $f$ on the closed interval $[0,1]$ such that

Let $\left\{ f_n\right\}_{n=1}^{\infty }$ be a uniformly bounded equicontinuous sequence of real-valued functions on the compact metric space $(X,d)$. Define the functions $g_n : X \to \mathbb {R}$, for $n \in \mathbb {N}$ by

Prove that the sequence $\left\{ g_n\right\}_{n=1}^{\infty }$ converges uniformly.

Let $\mathcal{F}$ be a uniformly bounded, equicontinuous family of real valued functions on the metric space $(X,d)$. Prove that the function

is continuous.

Let $X$ and $Y$ be metric spaces and $f$ a continuous map of $X$ into $Y$. Let $K_1, K_2, \ldots$ be nonempty compact subsets of $X$ such that $K_{n+1} \subset K_n$ for all $n$, and let $K = \bigcap K_n$. Prove that $f(K) = \bigcap f(K_n)$.

Let $(X_1, d_1)$ and $(X_2, d_2)$ be metric spaces and $f : X_1 \to X_2$ a continuous surjective map such that $d_1(p,q) \leq d_2(f(p), f(q))$ for every pair of points $p, q$ in $X_1$.