EACH REGULAR PARATOPOLOGICAL GROUP IS COMPLETELY REGULAR

被引：39

作者：

Banakh, Taras ^{[1
,2
]}

Ravsky, Alex ^{[3
]}

机构：

[1] Ivan Franko Natl Univ Lviv, Dept Math, UA-79000 Lvov, Ukraine

[2] Jan Kochanowski Univ, Kielce, Poland

[3] Natl Acad Sci, Pidstryhach Inst Appl Problems Mech & Math, UA-79060 Lvov, Ukraine

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2017年 / 145卷 / 03期

关键词：

Tychonoff space; regular space; completely regular space; semiregular space; Hausdorff space; functionally Hausdorff space; semi-Hausdorff space; separation axiom; quasi-uniformity; paratopological group; topological monoid; SEMIGROUPS;

D O I：

10.1090/proc/13318

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove that a semiregular topological space X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

引用

页码：1373 / 1382

页数：10

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