Hattori subspaces

For a subset A of an almost topological group $(G,\tau )$, 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 $(G_{⁎},{\tau }_{⁎})$ (respectively, $(G,\tau )$). Given an infinite subset X of an almost topological group G and $A\subseteq X$, we denote by $X\left(A\right)$, $X_{⁎}$ and X to the spaces $(X,\tau (A\left){|}_X\right)$, $(X,{\tau }_{⁎}{|}_X)$ and $(X,\tau {|}_X)$, respectively. We say that $X\left(A\right)$ is the Hattori subspace associated to A. In this paper, we obtain information about Hattori subspaces. We show that some known topological spaces can be obtained as Hattori subspaces of some almost topological groups.