A set s is compact if for every filter f that contains s, every set of f also meets every neighborhood of some a ∈ s.

Type class for compact spaces. Separation is sometimes included in the definition, especially in the French literature, but we do not include it here.

Tychonoff's theorem

There are various definitions of "locally compact space" in the literature, which agree for Hausdorff spaces but not in general. This one is the precise condition on X needed for the evaluation map C(X, Y) × X → Y to be continuous for all Y when C(X, Y) is given the compact-open topology.

A irreducible set is one where there is no non-trivial pair of disjoint opens.

A irreducible space is one where there is no non-trivial pair of disjoint opens.

A connected set is one where there is no non-trivial open partition.

A connected space is one where there is no non-trivial open partition.