The problem whether every topological space $X$ has a compactification $Y$ such that every continuous mapping $f$ from $X$ into a compact space $Z$ has a continuous extension from $Y$ into $Z$ is answered in the negative. For some spaces $X$ such compactifications exist.

Suppose that P is a finite 2-polyhedron. We prove that there exists a PL surjective map f:Q → P from a fake surface Q with preimages of f either points or arcs or 2-disks. This yields a reduction of the Whitehead asphericity conjecture (which asserts that every subpolyhedron of an aspherical 2-polyhedron is also aspherical) to the case of fake surfaces. Moreover, if the set of points of P having a neighbourhood homeomorphic to the 2-disk is a disjoint union of open 2-disks, and every point of P...

We study relations between the cellularity and index of narrowness in topological groups and their $G_{\delta }$-modifications. We show, in particular, that the inequalities $in\left({\left(H\right)}_{\tau }\right)\le 2^{\tau ·in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau }\right)\le 2^{2^{\tau ·in\left(H\right)}}$ hold for every topological group $H$ and every cardinal $\tau \ge \omega$, where ${\left(H\right)}_{\tau }$ denotes the underlying group $H$ endowed with the $G_{\tau }$-modification of the original topology of $H$ and $in\left(H\right)$ is the index of narrowness of the group $H$. Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $\omega$-narrow group $G$ understood...

Given a discrete group $G$, we consider the set $ℒ\left(G\right)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ\left(G\right)$ into the Cantor cube ${\{0,1\}}^G$. We show that the cellularity $c\left(ℒ\left(G\right)\right)\le {\aleph }_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c\left(ℒ\right(G\left)\right)=\tau$.

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $\theta ={\left(2^{cf\left(\mu \right)}\right)}^{+}$ and $2^{\mu }={\mu }^{+}$ then there are Boolean algebras ${𝔹}_1,{𝔹}_2$ such that $c\left({𝔹}_1\right)=\mu ,c\left({𝔹}_2\right)<\theta butc({𝔹}_1*{𝔹}_2)={\mu }^{+}$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $𝔹$ is a ccc Boolean algebra and ${\mu }^{{\beth }_{\omega }}\le \lambda =cf\left(\lambda \right)\le 2^{\mu }$ then $𝔹$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.

A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case, every open set...

A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f:X\to ℝ$ upper bounded by the diameter of $X$ and any finite subset $B$ of $X$ there is $x\in X$ such that $f\left(a\right)=d(x,a)$ for each $a\in A\cup B$. Central subsets of the Urysohn universal space $𝕌$ (see introduction) are studied. It is proved that a metric space $X$ is isometrically embeddable into $𝕌$ as a central set iff $X$ has the collinearity property. The Katětov maps of the real line are characterized.