Compact covers and function spaces

Abstract

Abstract. For a Tychonoff space X, we denote by C p (X) and C c (X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively.Providing a characterization of the Lindelöf Σ-property of X in terms of C p (X), we extend Okunev's results by showing that if there exists a surjection from) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c (X) onto C c (Y ), then Y is a separable completely metrizable space. We study also the non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by p -equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and p -equivalent to a first countable Y , then Y is metrizable (V. Valov). Some additional results are presented.

Domination in products

“…In [8] the spaces (strongly) dominated by M = D(κ) ω are referred to as spaces having a P(κ)-compact resolution (swallowing compact sets).…”

“…We can show that this is not so by presenting the following theorem that first appeared as an external contribution of the author of this note to the paper [8]. Proof.…”

Spaces with an M-diagonal

“…Quite a few results have been published recently that address the subject as Basile and Darji wrote in [3]: "In recent years much attention has been enjoyed by topological spaces which are dominated by second countable spaces". To this regard, we refer the reader to the work of Gartside and Mamatelashvilli [11], also the paper by Islas and Jardón [18], and that of Kakol et al [21], besides the mentioned paper [3].…”