Small compacts

Abstract

For a subset A of an almost topological group G, the Hattori space $H\left(A\right)$ is a topological space whose underlying set is G and whose topology $\tau \left(A\right)$ is defined as follows: if $x\in A$ (respectively, $x\notin A$), then the neighborhoods of x in $H\left(A\right)$ are the same neighborhoods of x in the reflection group (respectively, G). In this paper, we show the following:

• i)
  G is an almost topological group if and only if the Hattori topology $\tau \left(A\right)$ can be defined on G for each subset A of G.
• ii)
  If A is a subset of a proper almost topological group G, then $H\left(A\right)$ is locally compact if and only if $G_{⁎}$ is locally compact, $G\setminus A$ is closed in $G_{⁎}$ and for each $x\in G\setminus A$, there exists $U\in {\beta }_e$ such that ${\overline{Ux}}^{{\tau }_{⁎}}\cap (G\setminus A)=\left\{x\right\}$ and ${\overline{Ux}}^{{\tau }_{⁎}}\setminus Vx$ is closed in $G_{⁎}$, for each $V\in {\beta }_e$.
• iii)
  If A is a subset of a proper almost topological group G such that $G_{⁎}$ has countable pseudocharacter, then $H\left(A\right)$ has small compacts if and only if A has small compacts.

Moreover, we study the property of being σ-compact in Hattori spaces $H\left(A\right)$, where A is a subset of an almost topological group G. We show that if G is a proper almost topological group, then G is σ-compact if and only if G is countable.