Hattori topologies on almost topological groups

Abstract

For a subset A of an almost topological group G, we define the Hattori topological space H(A), where H(A) is a topological space whose underlying set is G and whose topology is defined as follows: if x∈A (respectively, x∉A), then the neighborhoods of x in H(A) are the same neighborhoods of x in the reflection group (respectively, G). In this paper, we show that if G is an almost topological group and A is a proper subset of G, then H(A) is regular if and only if G is regular. We also prove that χ(H(A))=χ(G) for each proper subset A of G. If G is an almost topological group and G is not a topological group, we show the following:i)For each infinite subspace B of G, we have that nω(B)=|B|.ii)If A is a proper subset of G, then ω(H(A))=d(G)⋅χ(G)⋅|G∖A|.iii)In particular, if A is a proper subset of G, then H(A) is second-countable if and only if G is first-countable separable and G∖A is countable.iv)If A is a subset of G, then nω(H(A))=nω(G⁎)⋅(|G∖A|+ω).