On Hattori spaces

Abstract

For a subset $A$ of the real line $\mathbb R$, Hattori space $H(A)$ is a topological space whose underlying point set is the reals $\mathbb R$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H(A)$ to be respectively almost Cech-complete, Cech-complete, quasicomplete, Cech-analytic and weakly separated (in Tkacenko sense). Some of these results solve questions raised by V.A. Chatyrko and Y. Hattori.