We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

A space X is called an α-Toronto space if X is scattered of Cantor-Bendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprāns by constructing a countable α-Toronto space for each α ≤ ω. We also construct consistent examples of countable α-Toronto spaces for each $\alpha <{\omega }_1$.

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a $G_{\delta }$. In addition some nonperfect spaces with σ-disjoint bases are constructed.

For a transfinite cardinal κ and i ∈ 0,1,2 let ${ℒ}_i\left(\kappa \right)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $\kappa <2^{\aleph ₀}$, and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The class ℒ₁(κ)...

In this paper, we discuss covering properties in countable products of Čech-scattered spaces and prove the following: (1) If $Y$ is a perfect subparacompact space and $\{X_n:n\in \omega \}$ is a countable collection of subparacompact Čech-scattered spaces, then the product $Y\times {\prod }_{n\in \omega }{X}_n$ is subparacompact and (2) If $\{X_n:n\in \omega \}$ is a countable collection of metacompact Čech-scattered spaces, then the product ${\prod }_{n\in \omega }{X}_n$ is metacompact.

2000 Mathematics Subject Classification: 54C10, 54D15, 54G12.For given completely regular topological spaces X and Y, there is a completely regular space X ~⊗ Y such that for any completely regular space Z a mapping f : X × Y ⊗ Z is separately continuous if and only if f : X ~⊗ Y→ Z is continuous. We prove a necessary condition of normality, a sufficient condition of collectionwise normality, and a criterion of normality of the products X ~⊗ Y in the case when at least one factor is scattered.

It is shown that for every n ≥ 2 there exists an n-dimensional locally connected Polish space with Dimensionsgrad 1.

Let $X$ be a Hausdorff space and let $\mathscr{H}$ be one of the hyperspaces $CL\left(X\right)$, $𝒦\left(X\right)$, $ℱ\left(X\right)$ or ${ℱ}_n\left(X\right)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathscr{H}$: extremal disconnectedness, being a $F^{\text{'}}$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $𝒦\left(X\right)$ is hereditarily disconnected. We also...

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces,...

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each ${aₙ}_{n\in \mathbb{N}}conA⁺$ there are ${\lambda ₙ}_{n\in \mathbb{N}}\subseteq (0,\infty )$ and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...