Ordinal Compactness

Abstract

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities.

The most general form depends on two ordinal parameters.

Ordinal compactness turns out to be a much more varied notion than cardinal compactness.

We prove many nontrivial results of the form ``every $[ \alpha , \beta ]$-compact topological space is $[ \alpha', \beta ' ]$-compact'', for ordinals $\alpha$, $ \beta $, $\alpha'$ and $ \beta' $, while only trivial results of the above form hold, if we restrict to regular cardinals.

Counterexamples are provided showing that these results are optimal.

Many spaces satisfy the very same cardinal compactness properties, but have a broad range of distinct behaviors, as far as ordinal compactness is concerned.

A much more refined theory is obtained for $T_1$ spaces, in comparison with arbitrary topological spaces.

The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.