We show that MA${}_{\sigma -centered}\left({\omega }_1\right)$ implies that normal locally compact metacompact spaces are paracompact, and that MA(${\omega }_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

We prove that the one-point Lindelöfication of a discrete space of cardinality ω 1 is homeomorphic to a subspace of C p (X) for some hereditarily Lindelöf space X if the axiom [...] holds.

Two compact spaces are co-absoluteif their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω and βℝ are not co-absolute.

We construct from ⋄ a T₂ example of a hereditarily Lindelöf space X that is not a D-space but is the union of two subspaces both of which are D-spaces. This answers a question of Arhangel'skii.

We show that the small cardinal number $i=min\left\{\right|𝒜|:𝒜$ is a maximal independent family} has the following topological characterization: $i=min\{\kappa \le c:{\{0,1\}}^{\kappa }$ has a dense irresolvable countable subspace}, where ${\{0,1\}}^{\kappa }$ denotes the Cantor cube of weight $\kappa$. As a consequence of this result, we have that the Cantor cube of weight $c$ has a dense countable submaximal subspace, if we assume (ZFC plus $i=c$), or if we work in the Bell-Kunen model, where $i={\aleph }_1$ and $c={\aleph }_{{\omega }_1}$.

We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.

It is shown that no generalized Luzin space condenses onto the unit interval and that the discrete sum of ${\aleph }_1$ copies of the Cantor set consistently does not condense onto a connected compact space. This answers two questions from [2].

We consider the spaces called $Seq\left(u_t\right)$, constructed on the set $Seq$ of all finite sequences of natural numbers using ultrafilters $u_t$ to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that $S\left(u_t\right)$ is homogeneous if and only if all the ultrafilters $u_t$ have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to $Seq\left(p\right)$ (i.e., $u_t=p$ for all $t\in Seq$). It follows that for a Ramsey ultrafilter $p$, $Seq\left(p\right)$ is a topological group....

A function is two-to-one if every point in the image has exactly two inverse points. We show that every two-to-one continuous image of ℕ* is homeomorphic to ℕ* when the continuum hypothesis is assumed. We also prove that there is no irreducible two-to-one continuous function whose domain is ℕ* under the same assumption.

Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\mathbb{N}$ is a Lindelöf space, (2) $\mathbb{Q}$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:ℝ\to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8)...