Showing changes from revision #3 to #4: Added | Removed | Changed

Contents

Idea

The classical Heine–Borel theorem identifies those topological subspaces of Cartesian spaces that are compact in terms of simpler properties. A generalisation apples to all metric spaces and even to uniform spaces.

Versions

This is the classical theorem:

It's easy to prove that $S$ is closed precisely if it is a complete metric space as with the induced metric, and similarly $S$ is bounded precisely if it is totally bounded. This gives the next version:

This refers entirely to $S$ as a metric space in its own right. In fact it holds much more generally than for subspaces of a cartesian space:

This theorem refers only to uniform properties of $S$, and in fact a further generalistion is true:

The hard part is proving that a complete totally bounded space is compact; the converse is easy.

We could also try to generalise Theorem 1 to subspaces of other metric spaces, but this fails: every compact subspace of a metric space is closed and bounded (which is the easy direction), but not conversely.

Logical status

In the old days, one called a closed and bounded interval in the real line ‘compact’; once closedness and boundedness were generalised from intervals to arbitrary subsets (of the real line), the definition of ‘compact’ also generalised. The content of the theorem, then, is that this condition is equivalent to the modern definition of ‘compact’ using open covers, and indeed the definition was only derived afterwards as a name for the conclusion of the theorem.

In constructive mathematics, one sees several definitions of ‘compact’, which may make the theorem provable, refutable, or undecidable in various constructive systems. In intuitionism, Theorems 1 and 2 can be proved (using the fan theorem), but Theorems 3 and 4 cannot, leading Brouwer to define ‘compact’ (for a metric space) to mean complete and totally bounded. In other literature, one sometimes sees the abbreviation ‘CTB’ used instead. In Russian constructivism, already Theorems 1 and 2 can be refuted, but CTB spaces are still important.

In locale theory and other approaches to pointless topology, the open-cover definition of ‘compact’ is clearly correct, and the failure of CTB spaces to be compact (constructively) may be seen as a consequence of working with points. Already in Bishop's weak system of constructivism, every CTB metric space $X$ gives rise to a compact locale, which classically (assumingexcluded middle and dependent choice) is the locale of open subsets of $X$, but constructively requires a more nuanced construction. (I need to find the reference for this, which is by Douglas Bridges et al.)

References

According to Wikipedia, the theorem was first proved by Pierre Cousin in 1895. It is named after Eduard Heine (who used it but did not prove it) and Émile Borel (who proved a limited version of it), an instance ofBaez's law.

A proof is spelled out for instance at

• ProofWiki, Heine-Borel Theorem (General case), (Special case)