On quasi-uniform space valued semi-continuous functions

Kubiak, Tomasz, and Vicente, María Angeles de Prada. "On quasi-uniform space valued semi-continuous functions." Commentationes Mathematicae Universitatis Carolinae 50.1 (2009): 125-133. <http://eudml.org/doc/32486>.

@article{Kubiak2009,
abstract = {F. van Gool [Comment. Math. Univ. Carolin. 33 (1992), 505–523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space $(R,\mathcal \{U\})$. 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 $T(\mathcal \{U\})$ generated by the quasi-uniformity $\mathcal \{U\}$, so that many of his preparatory results become consequences of standard topological facts. In particular, when the order induced by $\mathcal \{U\}$ makes $R$ into a continuous lattice, then $T(\mathcal \{U\})$ agrees with the Scott topology $\sigma (R)$ on $R$ and, thus, the lower semicontinuity reduces to a well known concept.},
author = {Kubiak, Tomasz, Vicente, María Angeles de Prada},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {lower semi-continuity; quasi-uniformity; continuous lattice; lower semi-continuity; quasi-uniformity; continuous lattice},
language = {eng},
number = {1},
pages = {125-133},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On quasi-uniform space valued semi-continuous functions},
url = {http://eudml.org/doc/32486},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Kubiak, Tomasz
AU - Vicente, María Angeles de Prada
TI - On quasi-uniform space valued semi-continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 1
SP - 125
EP - 133
AB - F. van Gool [Comment. Math. Univ. Carolin. 33 (1992), 505–523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space $(R,\mathcal {U})$. 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 $T(\mathcal {U})$ generated by the quasi-uniformity $\mathcal {U}$, so that many of his preparatory results become consequences of standard topological facts. In particular, when the order induced by $\mathcal {U}$ makes $R$ into a continuous lattice, then $T(\mathcal {U})$ agrees with the Scott topology $\sigma (R)$ on $R$ and, thus, the lower semicontinuity reduces to a well known concept.
LA - eng
KW - lower semi-continuity; quasi-uniformity; continuous lattice; lower semi-continuity; quasi-uniformity; continuous lattice
UR - http://eudml.org/doc/32486
ER -