Precompact groups have accessible boundaries of open sets
Keywords:
accessible boundary, precompact group, $\lambda$-accessible boundary, tightness, sequential locally convex space, pseudocompact group, locally pseudocompact group, $\kappa$-Fréchet–Urysohn propertyAbstract
Given an infinite cardinal $\lambda$, a space $X$ has $\lambda$-accessible boundaries of its open sets if $\overline{U} = \bigcup \{\overline{A} : A \in [U]^{\leq \lambda}\}$ for every open set $U \subset X$. If $X$ has the above-mentioned property for $\lambda = \omega$, then $X$ is said to have accessible boundaries of open sets. We show that every precompact topological group has accessible boundaries of open sets. As a consequence, every locally pseudocompact topological group has accessible boundaries of open sets. If a topological group $G$ is either locally compact or $\omega$-bounded, then $G$ turns out to be $\kappa$-Fréchet–Urysohn. Our results solve two published open questions.
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