We show that if $X$ is a subspace of a linearly ordered space, then $C_k\left(X\right)$ is a Baire space if and only if $C_k\left(X\right)$ is Choquet iff $X$ has the Moving Off Property.

A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $ℬ$ such that every base ${ℬ}^{\text{'}}\subseteq ℬ$ has a locally finite subcover $𝒞\subseteq {ℬ}^{\text{'}}$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.

Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g:X\to {𝕀}^k$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open.  Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g:X\to {𝕀}^k$ such that dim (f × g) = 1. We improve this result of Sternfeld showing...

Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.

The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{\alpha }}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left(X_{\alpha }\cap X_{\beta }\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${X_{\alpha }}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_G{X}_{\alpha }=n$ for all α ∈ J, then $di{m}_G\left(X_{\alpha }\cap X_{\beta }\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...