@article{Ferrando2024, 
author = {Juan C. Ferrando and Manuel López-Pellicer and Santiago Moll-López},
title = {Covering properties of        C          p            (          Y              |            X        )},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {7},
pages = {17743-17757},
keywords = {Lindelöf Σ-space, Polish space, P-space, bounded resolution, bornological space, Banach disk, quasi-(LB)-space},
url = {https://www.sciopen.com/article/10.3934/math.2024862},
doi = {10.3934/math.2024862},
abstract = {Let    X be an infinite Tychonoff space, and    Y be a topological subspace of    X. In this paper, we study some covering properties of the subspace        C          p            (          Y              |            X        )   of        C          p            (    Y    )   consisting of those functions    f  ∈  C      (    Y    )   which admit a continuous extension to    X equipped with the relative topology of        C          p            (    Y    )  . Among other results, we show that        (    i    )          C          p        (  Y      |    X  ) has a fundamental bounded resolution if and only if    Y is countable; when    X is realcompact and    Y is closed in    X, we have        (          i      i        )   if        C          p        (  Y      |    X  ) admits a resolution of convex compact sets that swallows the local null sequences in        C          p        (  Y      |    X  ), then    Y is countable and discrete;        (          i      i      i        )   if        C          p        (  Y      |    X  ) admits a compact resolution that swallows the compact sets, then    Y is also countable and discrete, and, as a corollary, we deduce that        C          p        (  Y      |    X  ) admits a compact resolution that swallows the compact sets if and only if        C          p        (  Y      |    X  ) is a Polish space. We also prove that        (          i      v        )   for a metrizable space    X,        C          p            (    X    )   is a quasi-       (          L      B        )  -space if and only if    X is    σ-compact, and hence for a subspace    Y of    X, the space        C          p            (          Y              |            X        )   is a quasi-       (          L      B        )  -space. We include some examples and observations that answer natural questions raised in this paper.}
}