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 \v Cech-complete, \v Cech-complete, quasicomplete, \v Cech-analytic and weakly separated (in Tkacenko sense). Some of these results solve questions raised by V.A. Chatyrko and Y. Hattori.
Keywords: Hattori space; \v Cech-complete space; \v Cech-analytic space; neighborhood assignment; Sorgenfrey line; scattered set; weakly separated space
DOI: DOI 10.14712/1213-7243.2015.199
AMS Subject Classification: 54C05 54C35 54C45 54C99