Heine-Cantor Theorem/Proof 1

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Theorem

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $M_1$ be compact.

Let $f: A_1 \to A_2$ be a continuous mapping.


Then $f$ is uniformly continuous.


Proof

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.


For all $x \in A_1$, define:

$\map \Delta x = \set {\delta \in \R_{>0}: \forall y \in A_1: \map {d_1} {x, y} < 2 \delta \implies \map {d_2} {\map f x, \map f y} < \dfrac \epsilon 2}$

Define:

$\mathcal C = \set {\map {B_{\delta} } x: x \in A_1, \, \delta \in \map \Delta x}$

where $B_{\delta} \left({x}\right)$ denotes the open $\delta$-ball of $x$ in $M_1$.

From the definition of continuity, it follows that $\mathcal C$ is a cover for $A_1$.

From Open Ball of Metric Space is Open Set, it therefore follows that $\mathcal C$ is an open cover for $A_1$.


By the definition of a compact metric space, there exists a finite subcover $\set {\map {B_{\delta_1} } {x_1}, \map {B_{\delta_2} } {x_2}, \ldots, \map {B_{\delta_n} } {x_n} }$ of $\mathcal C$ for $A_1$.

Define:

$\delta = \min \set {\delta_1, \delta_2, \ldots, \delta_n}$


Let $x, y \in A_1$ satisfy $\map {d_1} {x, y} < \delta$.

By the definition of a cover, there exists a $k \in \set{1, 2, \ldots, n}$ such that $\map {d_1} {x, x_k} < \delta_k$.

Then:

\(\displaystyle \map {d_1} {y, x_k}\) \(\le\) \(\displaystyle \map {d_1} {y, x} + \map {d_1} {x, x_k}\) Triangle Inequality
\(\displaystyle \) \(<\) \(\displaystyle \delta + \delta_k\) Metric Space Axiom $\paren {M3}$
\(\displaystyle \) \(\le\) \(\displaystyle 2 \delta_k\)


By the definition of $\Delta \left({x_k}\right)$, it follows that:

$\map {d_2} {\map f x, \map f {x_k} } < \dfrac \epsilon 2$
$\map {d_2} {\map f y, \map f {x_k} } < \dfrac \epsilon 2$

Hence:

\(\displaystyle \map {d_2} {\map f x, \map f y}\) \(\le\) \(\displaystyle \map {d_2} {\map f x, \map f {x_k} } + \map {d_2} {\map f {x_k}, \map f y}\) Triangle Inequality
\(\displaystyle \) \(<\) \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) Metric Space Axiom $\paren{M3}$
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\)

The result follows from the definition of uniform continuity.

$\blacksquare$


Source of Name

This entry was named for Heinrich Eduard Heine and Georg Cantor.


Sources