Local properties on the remainders of the topological groups

TL;DR

This paper investigates conditions under which the remainder of a topological group’s compactification is locally characterized by certain base properties, leading to the group and its compactification being separable and metrizable.

Contribution

It improves existing results by establishing new local base conditions on the remainder that guarantee separability and metrizability of the group and its compactification.

Findings

If the remainder has a locally point-countable p-metabase and countable π-character, then the group and compactification are separable and metrizable.

Having a locally δθ-base on the remainder implies the group and compactification are separable and metrizable.

A locally quasi-Gδ-diagonal on the remainder ensures the group and compactification are separable and metrizable.

Abstract

When does a topological group $G$ have a Hausdorff compactification $bG$ with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A.V. Arhangel'ski\v{\i} and C. Liu's. Let $G$ be a non-locally compact topological group and $bG$ be a compactification of $G$. The following facts are established: (1) If $bG\setminus G$ has a locally a point-countable $p$-metabase and $\pi$-character of $bG\setminus G$ is countable, then $G$ and $bG$ are separable and metrizable; (2) If $bG\setminus G$ has locally a $\delta\theta$-base, then $G$ and $bG$ are separable and metrizable; (3) If $bG\setminus G$ has locally a quasi-$G_{\delta}$-diagonal, then $G$ and $bG$ are separable and metrizable. Finally, we give a partial answer for a question, which was posed by C. Liu in \cite{LC}.