We prove that every Tychonoff strongly monotonically monolithic star countable space is Lindelöf, which solves a question posed by O.T. Alas et al. We also use this result to generalize a metrization theorem for strongly monotonically monolithic spaces. At the end of this paper, we study the extent of star countable spaces with k-in-countable bases, k ∈ ℤ.

A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $ℬ$ such that every base ${ℬ}^{\text{'}}\subseteq ℬ$ has a locally finite subcover $𝒞\subseteq {ℬ}^{\text{'}}$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.

We prove that if $X^n$ is a union of $n$ subspaces of pointwise countable type then the space $X$ is of pointwise countable type. If $X^{\omega }$ is a countable union of ultracomplete spaces, the space $X^{\omega }$ is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].

We consider the class of decreasing (G) spaces introduced by Collins and Roscoe and address the question as to whether it coincides with the class of decreasing (A) spaces. We provide a partial solution to this problem (the answer is yes for homogeneous spaces). We also express decreasing (G) as a monotone normality type condition and explore the preservation of decreasing (G) type properties under closed maps. The corresponding results for decreasing (A) spaces are unknown.

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,...

Some relationships between $1$-sequence-covering maps and weak-open maps or sequence-covering $s$-maps are discussed. These results are used to generalize a result from Lin S., Yan P., Sequence-covering maps of metric spaces, Topology Appl. 109 (2001), 301–314.

Some results concerning spaces with countably weakly uniform bases are generalized for spaces with $n$-in-countable ones.

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. The behavior of $\pi$-metrizable spaces under certain types of mappings is studied. In particular we characterize strongly $d$-separable spaces as those which are the image of a $\pi$-metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a $\pi$-metrizable space under an open continuous mapping. A question posed by Arhangel’skii regarding if a $\pi$-metrizable topological group must be metrizable receives...

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a $G_{\delta }$-diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered...

It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space $C_p\left(X\right)$ has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible...

Generalizing a theorem of Oxtoby, it is shown that an arbitrary product of Baire spaces which are almost locally universally Kuratowski-Ulam (in particular, have countable-in-itself π-bases) is a Baire space. Also, partially answering a question of Fleissner, it is proved that a countable box product of almost locally universally Kuratowski-Ulam Baire spaces is a Baire space.

Some constructions of spaces with/without dense subspaces satisfying stronger separation axioms are presented.

Given a subbase $𝒮$ of a space $X$, the game $PO(𝒮,X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point $x_n\in X$ and a set $U_n\in 𝒮$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n:n\in ø\}$ have been made and the player $P$ wins if ${\bigcup }_{n\in ø}{U}_n=X$; otherwise $O$ is the winner. Since $PO(𝒮,X)$ is an evident modification of the well-known point-open game $PO\left(X\right)$, the primary line of research is to describe the relationship between $PO\left(X\right)$ and $PO(𝒮,X)$ for a given subbase $𝒮$. It turns out that, for any subbase $𝒮$, the player $P$ has a winning strategy...

We characterize spaces with $\sigma$-$n$-linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the $𝐜$-productivity of the property of having a $\sigma$-linked base.