The Menger and projective Menger properties of function spaces with the set-open topology

Rok, strany: 2019, 699 - 706

For a Tychonoff space $X$ and a family $λ$ of subsets of $X$, we denote by $C_λ(X)$ the space of all real-valued continuous functions on $X$ with the set-open topology. A Menger space is a topological space in which for every sequence of open covers $\mathcal{U}₁, \mathcal{U}₂,…$ of the space there are finite sets $\mathcal{F}₁\subset \mathcal{U}₁, \mathcal{F}₂\subset \mathcal{U}₂, …$ such that family $\mathcal{F}₁\cup \mathcal{F}₂\cup …$ covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space $C_λ(X)$. Our main results state that (1) $C_λ(X)$ is Menger if and only if $C_λ(X)$ is $σ$-compact; (2) $C_p(Y\vert X)$ is projective Menger if and only if $C_p(Y\vert X)$ is $σ$-pseudocompact where $Y$ is a dense subset of $X$.

ISO 690:

Osipov, A. 2019. The Menger and projective Menger properties of function spaces with the set-open topology. In Mathematica Slovaca, vol. 69, no.3, pp. 699-706. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0258

APA:

Osipov, A. (2019). The Menger and projective Menger properties of function spaces with the set-open topology. Mathematica Slovaca, 69(3), 699-706. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0258