期刊
TOPOLOGY AND ITS APPLICATIONS
卷 200, 期 -, 页码 19-43出版社
ELSEVIER SCIENCE BV
DOI: 10.1016/j.topol.2015.12.010
关键词
Quasi-uniform space; Rotund quasi-uniform space; Quasi-pseudometric; Left-subinvariant premetric; Topological monoid; Paratopological group
资金
- NCN [DEC-2012/07/D/ST1/02087]
We define a notion of a rotund quasi-uniform space and describe a new direct construction of a (right-continuous) quasi-pseudometric on a (rotund) quasi-uniform space. This new construction allows to give alternative proofs of several classical metrizability theorems for (quasi-)uniform spaces and also obtain some new metrizability results. Applying this construction to topological monoids with open shifts, we prove that the topology of any (semiregular) topological monoid with open shifts is generated by a family of (right-continuous) left-subinvariant quasi-pseudometrics, which resolves an open problem posed by Raysky in 2001. This implies that a topological monoid with open shifts is completely regular if and only if it is semiregular. Since each paratopological group is a topological monoid with open shifts these results apply also to paratopological groups. (C) 2016 Elsevier B.V. All rights reserved.
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