Editing Heine–Cantor theorem

{{Short description|Mathematical theorem}}
{{distinguish|Cantor's theorem}}{{No footnotes|date=April 2019}}
In [[mathematics]], the '''Heine–Cantor theorem''' states that a [[continuous function]] between two [[metric space]]s is uniformly continuous if its domain is [[Compact space|compact]].
The theorem is named after [[Eduard Heine]] and [[Georg Cantor]].

{{Math theorem
|name=Heine–Cantor theorem
|If <math>f \colon M \to N</math> is a [[continuous function]] between two [[metric space]]s <math>M</math> and <math>N</math>, and <math>M</math> is [[compact space|compact]], then <math>f</math> is [[uniformly continuous]].
}}
An important special case of the Cantor theorem is that every continuous function from a [[Closed interval|closed]] [[bounded set|bounded]] [[Interval (mathematics)|interval]] to the [[real number]]s is uniformly continuous.

{{Math proof
|title=Proof of Heine–Cantor theorem
|Suppose that <math>M</math> and <math>N</math> are two [[Metric space|metric spaces]] with metrics <math>d_M</math> and <math>d_N</math>, respectively. Suppose further that a function <math>f: M \to N</math> is continuous and <math> M </math> is compact. We want to show that <math>f</math> is [[Uniform continuity|uniformly continuous]], that is, for every positive real number <math>\varepsilon > 0</math> there exists a positive real number <math>\delta > 0</math> such that for all points <math>x, y</math> in the [[Domain of a function|function domain]] <math>M</math>, <math>d_M(x,y) < \delta</math> implies that <math>d_N(f(x), f(y)) < \varepsilon</math>.

Consider some positive real number <math>\varepsilon > 0</math>. By [[Continuous function#Continuous functions between metric spaces|continuity]], for any point <math>x</math> in the domain <math>M</math>, there exists some positive real number <math>\delta_x > 0</math> such that <math>d_N(f(x),f(y)) < \varepsilon/2</math> when <math>d_M(x,y) < \delta _x</math>, i.e., a fact that <math>y</math> is within <math>\delta_x</math> of <math>x</math> implies that <math>f(y)</math> is within <math>\varepsilon / 2</math> of <math>f(x)</math>. 

Let <math>U_x</math> be the [[Open set|open]] <math>\delta_x/2</math>-neighborhood of <math>x</math>, i.e. the [[Set (mathematics)|set]]

:<math>U_x = \left\{ y \mid d_M(x,y) < \frac 1 2 \delta_x \right\}.</math>

Since each point <math>x</math> is contained in its own <math>U_x</math>, we find that the collection <math>\{U_x \mid x \in M\}</math> is an open [[Cover (topology)|cover]] of <math>M</math>. Since <math>M</math> is compact, this cover has a finite subcover <math>\{U_{x_1}, U_{x_2}, \ldots, U_{x_n}\}</math> where <math>x_1, x_2, \ldots, x_n \in M</math>. Each of these open sets has an associated radius <math>\delta_{x_i}/2</math>. Let us now define <math>\delta = \min_{1 \leq i \leq n} \delta_{x_i}/2</math>, i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this minimum <math>\delta</math> is well-defined and positive. We now show that this <math>\delta</math> works for the definition of uniform continuity.

Suppose that <math>d_M(x,y) < \delta</math> for any two <math>x, y</math> in <math>M</math>. Since the sets <math>U_{x_i}</math> form an open (sub)cover of our space <math>M</math>, we know that <math>x</math> must lie within one of them, say <math>U_{x_i}</math>. Then we have that <math>d_M(x, x_i) < \frac{1}{2}\delta_{x_i}</math>. The [[Triangle_inequality#Metric_space|triangle inequality]] then implies that

:<math>d_M(x_i, y) \leq d_M(x_i, x) + d_M(x, y) < \frac{1}{2} \delta_{x_i} + \delta \leq \delta_{x_i},</math>

implying that <math>x</math> and <math>y</math> are both at most <math>\delta_{x_i}</math> away from <math>x_i</math>. By definition of <math>\delta_{x_i}</math>, this implies that <math>d_N(f(x_i),f(x))</math> and <math>d_N(f(x_i), f(y))</math> are both less than <math>\varepsilon/2</math>. Applying the triangle inequality then yields the desired

:<math>d_N(f(x), f(y)) \leq d_N(f(x_i), f(x)) + d_N(f(x_i), f(y)) < \frac{\varepsilon}{2}  + \frac{\varepsilon}{2} = \varepsilon.</math>
[[Q.E.D.|∎]]}}

For an alternative proof in the case of <math>M = [a, b]</math>, a closed interval, see the article [[Non-standard calculus#Heine.E2.80.93Cantor theorem|Non-standard calculus]].

== See also ==

* [[Cauchy-continuous function]]

==External links==
* {{planetmath|urlname=heinecantortheorem|title=Heine–Cantor theorem}} 
* {{planetmath|urlname=proofofheinecantortheorem|title=Proof of Heine–Cantor theorem}}

{{DEFAULTSORT:Heine-Cantor theorem}}
[[Category:Theory of continuous functions]]
[[Category:Metric geometry]]
[[Category:Theorems in mathematical analysis]]
[[Category:Articles containing proofs]]