CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The classical Heine–Borel theorem identifies those topological subspaces of Cartesian spaces ($\mathbb{R}^n$s) that are compact in terms of simpler properties. A generalisation applies to all metric spaces and even to uniform spaces.
This is the classical theorem:
Let $S$ be a topological subspace of a Cartesian space ($S \subset \mathbb{R}^n$). Then $S$ is a compact topological space (with the induced topology) precisely if it is closed and bounded in $\mathbb{R}^n$.
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:
Let $S$ be a topological subspace of a Cartesian space ($S \subset \mathbb{R}^n$). Then $S$ is a compact topological space (with the induced topology) precisely if it is complete and totally bounded (with the induced metric?).
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:
Let $S$ be a metric space. Then $S$ is compact precisely if it is complete and totally bounded.
This theorem refers only to uniform properties of $S$, and in fact a further generalistion is true:
Let $S$ be a uniform space. Then $S$ is compact precisely if it is complete and totally bounded.
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.
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 Theorem 1, then, is that this condition is equivalent to the modern definition of ‘compact’ using open covers, and indeed the modern 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. Using the open-cover definition of ‘compact’, Theorems 1 and 2 are equivalent to the fan theorem (and so hold in intuitionism), but Theorems 3 and 4 are stronger and Brouwer could not prove them, leading him 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 using the open-cover definition, 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 (assuming excluded middle and dependent choice) is the locale of open subsets of $X$ but constructively requires a more nuanced construction; see Vickers.
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 of Baez's law.
A proof is spelled out for instance at
On constructing a compact locale from a CTB metric space: