The body of this paper falls into two independent sections. The first deals with the existence of cross-sections in $F_{\sigma }$-decompositions. The second deals with the extensions of the results on accessibility in the plane.

We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu$ such that $X^{\mu }$ can be condensed onto a normal ($\sigma$-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu$ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^{\mu }$, $\mu \le \nu$, contains a closed copy...

A topological space is non-separably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a non-separably connected complete metric space X under a monotone quotient map. The metric $d_X$ of the space X is economical in the sense that for each infinite subspace A ⊂ X the cardinality of the set $d_X(a,b):a,b\in A$ does not exceed the density of A, $|d_X\left(A\times A\right)|\le dens\left(A\right)$. The construction of the space X determines a functor : Top...

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

Let $X$ be a metric continuum. Let $F_n\left(X\right)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_n\left(X\right)$ provided that the following implication holds: if $Y$ is a continuum and $F_n\left(X\right)$ is homeomorphic to $F_n\left(Y\right)$, then $X$ is homeomorphic to $Y$. In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_2\left(X\right)$, and (2) let $X$ be an arcwise connected...

It is shown that there exists a $\sigma$-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma$ of $Y$ by dyadic compacta such that $\left|\gamma \right|\le 2^{\omega }$. We also show that if a homogeneous compact space $Y$ is a continuous...

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and...