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Title: On quasi-uniform space valued semi-continuous functions (English)
Author: Kubiak, Tomasz
Author: Vicente, María Angeles de Prada
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 50
Issue: 1
Year: 2009
Pages: 125-133
Summary lang: English
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Category: math
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Summary: F. van Gool [Comment. Math. Univ. Carolin. {\bf 33} (1992), 505--523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space $(R,\Cal U)$. This note provides a purely topological view at the basic ideas of van Gool. The lower semicontinuity of van Gool appears to be just the continuity with respect to the topology $T(\Cal U)$ generated by the quasi-uniformity $\Cal U$, so that many of his preparatory results become consequences of standard topological facts. In particular, when the order induced by $\Cal U$ makes $R$ into a continuous lattice, then $T(\Cal U)$ agrees with the Scott topology $\sigma (R)$ on $R$ and, thus, the lower semicontinuity reduces to a well known concept. (English)
Keyword: lower semi-continuity
Keyword: quasi-uniformity
Keyword: continuous lattice
MSC: 06B35
MSC: 54C08
MSC: 54E15
MSC: 54F05
idZBL: Zbl 1212.54043
idMR: MR2562809
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Date available: 2009-08-18T12:23:39Z
Last updated: 2013-09-22
Stable URL: http://hdl.handle.net/10338.dmlcz/133420
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