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Heine–Cantor theorem

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In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if is a continuous function between two metric spaces and , and is compact, then is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.


Suppose that and are two metric spaces with metrics and , respectively. Suppose further that a function is continuous and is compact. We want to show that is uniformly continuous, that is, for every positive real number there exists a positive real number such that for all points in the function domain , implies that .

Consider some positive real number . By continuity, for any point in the domain , there exists some positive real number such that when , i.e., a fact that is within of implies that is within of .

Let be the open -neighborhood of , i.e. the set

Since each point is contained in its own , we find that the collection is an open cover of . Since is compact, this cover has a finite subcover where . Each of these open sets has an associated radius . Let us now define , i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this minimum is well-defined and positive. We now show that this works for the definition of uniform continuity.

Suppose that for any two in . Since the sets form an open (sub)cover of our space , we know that must lie within one of them, say . Then we have that . The triangle inequality then implies that

implying that and are both at most away from . By definition of , this implies that and are both less than . Applying the triangle inequality then yields the desired

For an alternative proof in the case of , a closed interval, see the article Non-standard calculus.

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