### ${\aleph}_{1}$-directed inverse systems of continuous images of arcs.

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For every topological property $\mathcal{P}$, we define the class of $\mathcal{P}$-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 $\mathcal{P}$ for each $x\in X$. It is shown that every $\mathcal{P}$-approximable compact space has $\mathcal{P}$, if $\mathcal{P}$ 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 ${x}^{\text{'}}$: ${E}_{\epsilon}\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla}^{\text{'}}\xb7{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla}^{\text{'}}{|}^{-1/2}{\nabla}^{\text{'}}\xb7{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. 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 ${\nabla}^{\text{'}}\xb7{m}^{\text{'}}=0$, 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 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.