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 |
. |
Category:
|
math |
. |
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 |
. |
Date available:
|
2009-08-18T12:23:39Z |
Last updated:
|
2013-09-22 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133420 |
. |
Reference:
|
[1] Fletcher P., Lindgren W.F.: Quasi-uniform Spaces.Marcel Dekker, New York, 1982. Zbl 0583.54017, MR 0660063 |
Reference:
|
[2] Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S.: A Compendium of Continuous Lattices.Springer, Berlin, Heidelberg, New York, 1980. Zbl 0452.06001, MR 0614752 |
Reference:
|
[3] Gierz G., Lawson J.D.: Generalized continuous and hypercontinuous lattices.Rocky Mountain J. Math. 11 (1981), 271--296. Zbl 0472.06014, MR 0619676, 10.1216/RMJ-1981-11-2-271 |
Reference:
|
[4] van Gool F.: Lower semicontinuous functions with values in a continuous lattice.Comment. Math. Univ. Carolin. 33 (1992), 505--523. Zbl 0769.06005, MR 1209292 |
Reference:
|
[5] Liu Y.-M., Luo M.-K.: Lattice-valued mappings, completely distributive law and induced spaces.Fuzzy Sets and Systems 42 (1991), 43--56. Zbl 0739.54002, MR 1123576, 10.1016/0165-0114(91)90088-8 |
Reference:
|
[6] Murdeshwar M.G., Naimpally S.A.: Quasi-uniform Topological Spaces.Publ. P. Noordhoff Ltd., Groningen, 1966. Zbl 0139.40501, MR 0211386 |
Reference:
|
[7] Nachbin L.: Topology and Order.Van Nostrand Mathematical Studies, 24, Princeton, New Jersey, 1965. Zbl 0333.54002, MR 0219042 |
Reference:
|
[8] Page W.: Topological Uniform Structures.Dover, New York, 1989. Zbl 0734.46001, MR 1102896 |
Reference:
|
[9] Watson W.S.: M.R. 94j:54007.. |
Reference:
|
[10] Zhang De-Xue: Metrizable completely distributive lattices.Comment. Math. Univ. Carolin. 38 (1997), 137--148. MR 1455477 |
. |