Kuratowski's intersection theorem

Kuratowski's result is a generalisation of Cantor's intersection theorem.

The theorem is named for the Polish mathematician Kazimierz Kuratowski, who proved it in 1930.

Given a subset A ⊆ X, its Kuratowski measure of non-compactness α(A) ≥ 0 is defined by Note that, if A is itself compact, then α(A) = 0, since every cover of A by open balls of arbitrarily small diameter will have a finite subcover.

The converse is also true: if α(A) = 0, then A must be precompact, and indeed compact if A is closed.

In some sense, the quantity α(A) is a numerical description of "how non-compact" the set A is.