Compact Space - Other Forms of Compactness

Other Forms of Compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.

• Sequentially compact: Every sequence has a convergent subsequence.
• Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
• Pseudocompact: Every real-valued continuous function on the space is bounded.
• Limit point compact: Every infinite subset has an accumulation point.

While all these conditions are equivalent for metric spaces, in general we have the following implications:

• Compact spaces are countably compact.
• Sequentially compact spaces are countably compact.
• Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is given by 2, with the product topology (Scarborough & Stone 1966, Example 5.3).

A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version.

Another related notion which (by most definitions) is strictly weaker than compactness is local compactness.

Generalizations of compactness include H-closed and the property of being an H-set in a parent space. A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense. Whereas we say X is an H-set of Z if every cover of X with open sets of Z has a finite subfamily whose Z closure contains X.