The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{ch}$-extender (resp. $L_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous $L_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and...

An embedding from a Cartesian product of two spaces into the Cartesian product of two spaces is said to be factorwise rigid provided that it is the product of embeddings on the individual factors composed with a permutation of the coordinates. We prove that each embedding of a product of two pseudo-arcs into itself is factorwise rigid. As a consequence, if X and Y are metric continua with the property that each of their nondegenerate proper subcontinua is homeomorphic to the pseudo-arc, then X ×...

We effectively construct in the Hilbert cube $ℍ={[0,1]}^{\omega }$ two sets $V,W\subset ℍ$ with the following properties: (a) $V\cap W=\varnothing$, (b) $V\cup W$ is discrete-dense, i.e. dense in ${{[0,1]}_D}^{\omega }$, where ${[0,1]}_D$ denotes the unit interval equipped with the discrete topology, (c) $V$, $W$ are open in $ℍ$. In fact, $V={\bigcup }_{\mathbb{N}}{V}_i$, $W={\bigcup }_{\mathbb{N}}{W}_i$, where $V_i={\bigcup }_0^{2^{i-1}-1}{V}_{ij}$, $W_i={\bigcup }_0^{2^{i-1}-1}{W}_{ij}$. $V_{ij}$, $W_{ij}$ are basic open sets and $(0,0,0,...)\in V_{ij}$, $(1,1,1,...)\in W_{ij}$, (d) $V_i\cup W_i$, $i\in \mathbb{N}$ is point symmetric about $(1/2,1/2,1/2,...)$. Instead of $[0,1]$ we could have taken any $T_4$-space or a digital interval, where the resolution (number of points) increases with $i$.

Necessary conditions and sufficient conditions are given for $C_p\left(X\right)$ to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with $C_p\left(K\right)$ an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).

Given a space $X$, its $G_{\delta }$-subsets form a basis of a new space $X_{\omega }$, called the $G_{\delta }$-modification of $X$. We study how the assumption that the $G_{\delta }$-modification $X_{\omega }$ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_{\omega }$ is discrete and, hence, homogeneous. Thus, $X_{\omega }$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_{\delta }$-modification of $X$ is homogeneous, then the weight $w\left(X\right)$ of $X$ does not exceed $2^{\omega }$ (Theorem 1). We also establish...

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

We prove that if some power of a space X is rectifiable, then $X^{\pi w\left(X\right)}$ is rectifiable. It follows that no power of the Sorgenfrey line is a topological group and this answers a question of Arhangel’skiĭ. We also show that in Mal’tsev spaces of point-countable type, character and π-character coincide.

In this expository paper it is shown that Martin's Axiom and the negation of the Continuum Hypothesis imply that the product of ccc spaces is a ccc space. The Continuum Hypothesis is then used to construct the Laver-Gavin example of two ccc spaces whose product is not a ccc space.

We consider isometry groups of a fairly general class of non standard products of metric spaces. We present sufficient conditions under which the isometry group of a non standard product of metric spaces splits as a permutation group into direct or wreath product of isometry groups of some metric spaces.

The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.