A space $X$ is discretely absolutely star-Lindelöf if for every open cover $𝒰$ of $X$ and every dense subset $D$ of $X$, there exists a countable subset $F$ of $D$ such that $F$ is discrete closed in $X$ and $St(F,𝒰)=X$, where $St(F,𝒰)=\bigcup \{U\in 𝒰:U\cap F\ne \varnothing \}$. We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed subspace.

In [Fund. Math. 210 (2010), 1-46] we claimed the truth of two statements, one now known to be false and a second lacking a proof. In this "Errata" we report these matters in the interest of setting the record straight on the status of these claims.

The statement in the title solves a problem raised by T. Retta. We also present a variation of the result in terms of $[\mu ,\kappa ]$-compactness.

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable,...

It is well known that every $ℝ$-factorizable group is $\omega$-narrow, but not vice versa. One of the main problems regarding $ℝ$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega$-narrow group is a continuous homomorphic image of an $ℝ$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $ℝ$-factorizable...

We investigate how the Lindelöf property of the function space $C_p(X,Y)$ is influenced by slight changes in $X$ and/or $Y$.

It is shown that if $X$ is a first-countable countably compact subspace of ordinals then $C_p\left(X\right)$ is Lindelöf. This result is used to construct an example of a countably compact space $X$ such that the extent of $C_p\left(X\right)$ is less than the Lindelöf number of $C_p\left(X\right)$. This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

This work presents some cardinal inequalities in which appears the closed pseudo-character, ${\psi }_c$, of a space. Using one of them — ${\psi }_c\left(X\right)\le 2^{d\left(X\right)}$ for $T_2$ spaces — we improve, from $T_3$ to $T_2$ spaces, the well-known result that initially $\kappa$-compact $T_3$ spaces are $\lambda$-bounded for all cardinals $\lambda$ such that $2^{\lambda }\le \kappa$. And then, using an idea of A. Dow, we prove that initially $\kappa$-compact $T_2$ spaces are in fact compact for $\kappa =2^{F\left(X\right)}$, $2^{s\left(X\right)}$, $2^{t\left(X\right)}$, $2^{\chi \left(X\right)}$, $2^{{\psi }_c\left(X\right)}$ or $\kappa =max\{{\tau }^{+},{\tau }^{<\tau }\}$, where $\tau >t(p,X)$ for all $p\in X$.