The number of K-determination of topological spaces

Abstract

We introduce a cardinal function that assigns to each topological space Y a cardinal number ℓΣ (Y) that measures how the space is determined by its compact subsets via upper semicontinuous compact valued maps defined on metric spaces. By doing so we extend and take to a different dimension the study of the so-called countably K-determined spaces (or Lindelöf Σ-spaces) and their associates Gul’ko compacta. We study the behaviour of ℓΣ(·) with respect to the usual operations for topological spaces as well as some of the standard operations within the category of Banach spaces. We study the relationship of ℓΣ(·) with regard to other cardinal functions like for instance the weight w(·) of spaces, for which we observe that although for any compact space K we always have ${\ell\Sigma (C(K),\tau_p)\leq w (C(K),\tau_p)}$ there is a space ${\mathbb Y}$ such that ${w (\mathbb Y) < \ell\Sigma (\mathbb Y)}$ : the example ${\mathbb Y}$ is a subspace of ${\beta\mathbb{N}}$ of cardinality ${2^{2^{\omega}}}$ whose compact subsets are finite. We also study some weakening of G _δ -conditions for diagonal of compact spaces that still imply metrizability of the underlying space and that have numerous applications in functional analysis. We close the paper establishing the relationship between ℓΣ(·), the Σ-degree introduced by Hödel and the class of strong Σ-spaces studied by Nagami and others.