Let $P$ be a topological property. A space $X$ is said to be star $P$ if whenever $𝒰$ is an open cover of $X$, there exists a subspace $A\subseteq X$ with property $P$ such that $X=\mathrm{St}(A,𝒰)$, where $\mathrm{St}(A,𝒰)=\bigcup \{U\in 𝒰:U\cap A\ne \varnothing \}.$ In this paper, we study the relationships of star $P$ properties for $P\in \{\mathrm{Lindel}ö\mathrm{f},\mathrm{compact},\mathrm{countablycompact}\}$ in pseudocompact spaces by giving some examples.

We provide a further estimate on the cardinality of a power homogeneous space. In particular we show the consistency of the formula $\left|X\right|\le 2^{\pi \chi \left(X\right)}$ for any regular power homogeneous ccc space.

Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with ${\aleph }_1$ generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra...

Every crowded space $X$ is $\omega$-resolvable in the c.c.c. generic extension $V^{Fn\left(\right|X|,2)}$ of the ground model. We investigate what we can say about $\lambda$-resolvability in c.c.c. generic extensions for $\lambda >\omega$. A topological space is monotonically ${\omega }_1$-resolvable if there is a function $f:X\to {\omega }_1$ such that $$\{x\in X:f\left(x\right)\ge \alpha \}{\subset }^{dense}X$$ for each $\alpha <{\omega }_1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\omega }_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically ${\omega }_1$-resolvable; (3) $X$ is ${\omega }_1$-resolvable in the Cohen-generic extension $V^{Fn({\omega }_1,2)}$....

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

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...