Ordinal Compactness

TL;DR

This paper introduces a new form of compactness based on ordinal parameters, revealing diverse behaviors in topological spaces and refining understanding for T1 spaces compared to classical cardinal-based compactness.

Contribution

It defines ordinal compactness, explores its properties, and demonstrates varied behaviors across spaces, especially refining the theory for T1 spaces.

Findings

Spaces can share the same cardinal compactness but differ in ordinal compactness behaviors.

Ordinal compactness exhibits more complexity and variety than classical cardinal compactness.

The theory simplifies for spaces of small cardinality.

Abstract

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the particular case when the parameters are cardinal numbers, we get back a classical notion. Generalized to ordinal numbers, this notion turns out to behave in a much more varied way. We present many examples of spaces satisfying the very same cardinal compactness properties, but with a broad range of distinct behaviors, with respect to ordinal compactness. 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.