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...

The paper is devoted to the study of the ordered set $A\mathcal{K}\left(X,\alpha \right)$ of all, up to equivalence, $A$-compactifications of an Alexandroff space $\left(X,\alpha \right)$. The notion of $A$-weight (denoted by $aw\left(X,\alpha \right)$) of an Alexandroff space $\left(X,\alpha \right)$ is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of $A\mathcal{K}\left(X,\alpha \right)$ and $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ are studied, where $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ is the set of all, up to equivalence, $A$-compactifications $Y$ of $\left(X,\alpha \right)$ for which $w\left(Y\right)=aw\left(X,\alpha \right)$. A characterization of the families of bounded functions generating an $A$-compactification of $\left(X,\alpha \right)$ is obtained. The notion...

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...

We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence ${𝕃}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/\left(p\right)$. Further, we show that $(\mathbb{Z},{𝕃}_1^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.