Categorical properties of semi-continuous quasi-ordered spaces

Commun. Korean Math. Soc. 2002 Vol. 17, No. 4, 681-691 Printed December 1, 2002

Seon Ho Shin Sookmyung Women's University

Abstract : We study categorical properties of the category \linebreak {\scriptsize $\mathbf{SWQOS}$} $(\mathbf{S^UWQOS}$, $\mathbf{S^LWQOS}$, resp.) of (upper, lower, resp.) semi-continuous quasi-ordered spaces and the subcategory {\scriptsize$\mathbf{SWPOS}$} ($\mathbf{S^UWPOS}$, $\mathbf{S^LWPOS}$, resp.) of (upper, lower, resp.) semi-continuous partially ordered spaces. We show that the categories $\mathbf{S^UWPOS}$, $\mathbf{S^LWPOS}$ and $\mathbf{SWPOS}$ are closed under the formation of initial mono-sources in the category $\mathbf{TQOS}$ of topological quasi-ordered spaces, and they are mono-topological, complete and cocomplete epireflective subcategories of the category $\mathbf{TQOS}$. We obtain their MacNeille and universal initial completions, as well as those of subcategories $\mathbf{S^UWQOS (S^LWQOS)}$ and $\mathbf{SWQOS}$, in which the topologies are $T_0$ and $T_1$, respectively.