Affiliation:
1. Dipartimentum Topologiæ S. Mathematicæ, Viale della Ricerca Scientifica II Università di Roma (Tor Vergata), Rome, Italy
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 [?,?]-compact topological space is
[?',?']-compact?, for ordinals ?,?, ?'and ?' while only trivial results
of the above form hold, if we restrict to regular cardinals. Counterexamples
are provided showing that many 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 T1 spaces, in comparison with arbitrary
topological spaces. The notion of ordinal compactness becomes partly trivial
for spaces of small cardinality.
Publisher
National Library of Serbia