For every topological property $𝒫$, we define the class of $𝒫$-approximable spaces which consists of spaces X having a countable closed cover $\gamma$ such that the “section” $X(x,\gamma )=\bigcap \{F\in \gamma :x\in F\}$ has the property $𝒫$ for each $x\in X$. It is shown that every $𝒫$-approximable compact space has $𝒫$, if $𝒫$ is one of the following properties: countable tightness, ${\aleph }_0$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or $2^{{\aleph }_0}<2^{{\aleph }_1}$). Metrizable-approximable spaces are studied: every compact space in this class has...

We characterize Corson-compact spaces by means of countable elementary substructures.

We answer a question of I. Juhasz by showing that MA $+\neg$ CH does not imply that every compact ccc space of countable $\pi$-character is separable. The space constructed has the additional property that it does not map continuously onto $I^{{\omega }_1}$.

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are called T₁-complementary provided that there exists a bijection f: X → Y such that ${\tau }_X$ and $f^{-1}\left(U\right):U\in {\tau }_Y$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

We construct a Hausdorff topological group $G$ such that ${\aleph }_1$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$-embedded in $G$, but $G$ is not $ℝ$-factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.