We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $<𝔠$ constituted by infinite sets, (2) $\chi \left(X\right)<𝔠$, (3) $X$ is a $T_2$ Baire space and $c\left(X\right)\le {\aleph }_0$ and (4) $X$ is a $T_1$ Baire space and has a network $𝒩$ with cardinality $<𝔠$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of...

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomorphic to the measure algebra of $2^{{\omega }_1}$. The first construction uses ♢ to produce an S-space with no convergent sequences in which every perfect set is a $G_{\delta }$. A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The second space is constructed under CH and is both HS and HL.

A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X=A{\bowtie }_{x}B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...

We prove that if ℱ is a non-meager P-filter, then both ℱ and ${}^{\omega }ℱ$ are countable dense homogeneous spaces.

The following statement is proved to be independent from $[LB+\neg CH]$: $(*)$ Let $X$ be a Tychonoff space with $c\left(X\right)\le {\aleph }_0$ and $\pi w\left(X\right)<ℭ$. Then a union of less than $ℭ$ of nowhere dense subsets of $X$ is a union of not greater than $\pi w\left(X\right)$ of nowhere dense subsets.

We consider the families of all subspaces of size ω₁ of $2^{\omega ₁}$ (or of a compact zero-dimensional space X of weight ω₁ in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω₁-sequences. Various relations among these families modulo the club filter in ${\left[X\right]}^{\omega ₁}$ are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩ M for an elementary submodel M of size ω₁. Various results with this flavor are obtained. Another tool used...

A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^{n+1}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.

We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space $X$ whose square $X^2$ is again Lindelöf but its cube $X^3$ has a closed discrete subspace of size ${𝔠}^{+}$, hence the Lindelöf degree $L\left(X^3\right)={𝔠}^{+}$. In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space $X$ such that $L\left(X^n\right)={\aleph }_0$ for all positive integers $n$, but $L\left(X^{{\aleph }_0}\right)={𝔠}^{+}={\aleph }_2$.