Hemicompactness and the kω-property of free topological vector spaces
Abstract
For a Tychonoff space X we consider the vector space E(X) with basis X endowed with one of the following topologies: the weak topologies Lb(X) and Lp(X) generated by Cb(X) and C(X), the free locally convex space L(X), the free locally r-convex space L(r)(X) (0<r≤1), the free locally pseudoconvex space L(0)(X), and the free topological vector space V(X). We prove that Lb(X) is hemicompact if and only if X is finite; that for every vector topology between Lp(X) and L(0)(X) hemicompactness is equivalent to X being countable with all compact subsets finite; and that V(X), as well as the free topological groups F(X) and A(X), are hemicompact if and only if X is hemicompact. In particular, V([0,1]) is hemicompact, while L([0,1]) and Lp([0,1]) are not. We also describe when these spaces are kω-spaces: Lb(X) and Lp(X) only for finite X, and L(r)(X), 0≤ r≤1, only for countable discrete X.
Related
- Extending a property of vector spaces to abelian groups
- Universal approximation of continuous maps between topological vector spaces
- The Ball-Covering Property in Lipschitz-Free Spaces
- The Dunford-Pettis property for Lipschitz-free spaces over ℝn
- Metrizable and Fréchet--Urysohn subgroups and linear subspaces of topological vector spaces
BibTeX
@misc{reznichenko2026hemicompactness,
title = {{Hemicompactness and the $k_ω$-property of free topological vector spaces}},
author = {Evgenii Reznichenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23020},
eprint = {2610.23020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23020},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hemicompactness and the k_ω-property of free topological vector spaces
AU - Evgenii Reznichenko
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23020
UR - https://arxiv.org/abs/2610.23020
AB - For a Tychonoff space X we consider the vector space E(X) with basis X endowed with one of the following topologies: the weak topologies L_b(X) and L_p(X) generated by C_b(X) and C(X), the free locally convex space L(X), the free locally r-convex space L^(r)(X) (0<r≤1), the free locally pseudoconvex space L^(0)(X), and the free topological vector space V(X). We prove that L_b(X) is hemicompact if and only if X is finite; that for every vector topology between L_p(X) and L^(0)(X) hemicompactness is equivalent to X being countable with all compact subsets finite; and that V(X), as well as the free topological groups F(X) and A(X), are hemicompact if and only if X is hemicompact. In particular, V([0,1]) is hemicompact, while L([0,1]) and L_p([0,1]) are not. We also describe when these spaces are k_ω-spaces: L_b(X) and L_p(X) only for finite X, and L^(r)(X), 0≤ r≤1, only for countable discrete X.
ER -