Open AccessEach regular paratopological group is completely regular
Author(s) -
Тарас Банах,
Alex Ravsky
Publication year - 2016
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/proc/13318
Subject(s) - hausdorff space , countable set , group (periodic table) , mathematics , second countable space , discrete mathematics , computer science , algorithm , combinatorics , physics , quantum mechanics
We prove that a semiregular topological space X 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.