Combinatorics of open covers (IX): Basic properties

Abstract

We introduce the concepts of diagonalization basis property and strong diagonalization basis property. For appropriate spaces having these properties we show that the classical selection properties are equivalent to certain basis properties of the spaces. In particular, these equivalences hold for various metrizable spaces. The Sorgenfrey line, which is not metrizable, has the diagonalization basis property and thus our results also apply in this case. We calculate critical selection cardinals for subspaces of the Sorgenfrey line.