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Ordinal Compactness

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.