This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma$-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

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...

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.

A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction $r:X̃\genfrac{}{}{0pt}{}{onto}{⟶}X$ such that all fibers $r^{-1}\left(x\right)$ are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum $Y_X$ is called a continuous pseudo-fan of a compactum X if there are a point $c\in Y_X$ and a family ℱ of pseudo-arcs such that $\bigcup ℱ=Y_X$, any subcontinuum of $Y_X$ intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with...

We show that, under CH, the corona of a countable ultrametric space is homeomorphic to ${\omega }^{*}$. As a corollary, we get the same statements for the Higson’s corona of a proper ultrametric space and the space of ends of a countable locally finite group.

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.