On paratopological groups

TL;DR

This paper explores various properties of paratopological groups, constructing examples and proving theorems that clarify conditions for submetrizability, developability, and H-closedness, thereby answering longstanding questions in the field.

Contribution

It provides new examples of non-submetrizable and non-metrizable paratopological groups, and establishes key conditions under which these groups exhibit certain topological properties.

Findings

Constructed a non-submetrizable Hausdorff paratopological group where every point is a G_delta-set.

Proved that first-countable Abelian paratopological groups are submetrizable.

Showed the Sorgenfrey line is not H-closed, answering a question by Arhangel'skind Tkachenko.

Abstract

In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group $G$ in which every point is a $G_{\delta}$-set, which gives a negative answer to Arhangel'ski\v{\i}\ and Tkachenko's question [Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. We prove that each first-countable Abelian paratopological group is submetrizable. Moreover, we discuss developable paratopological groups and construct a non-metrizable, Moore paratopological group. Further, we prove that a regular, countable, locally $k_{\omega}$-paratopological group is a discrete topological group or contains a closed copy of $S_{\omega}$. Finally, we discuss some properties on non-H-closed paratopological groups, and show that Sorgenfrey line is not H-closed, which gives a negative answer to Arhangel'ski\v{\i}\ and Tkachenko's question [Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. Some questions are posed.