Lower semicontinuous functions with values in a continuous lattice

Frans Gool

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 3, page 505-523
  • ISSN: 0010-2628

Abstract

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It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.

How to cite

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Gool, Frans. "Lower semicontinuous functions with values in a continuous lattice." Commentationes Mathematicae Universitatis Carolinae 33.3 (1992): 505-523. <http://eudml.org/doc/247357>.

@article{Gool1992,
abstract = {It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.},
author = {Gool, Frans},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {continuous lattices; lower semicontinuous functions; potential theory; continuous lattice; semiuniform structure; order; Lawson topology},
language = {eng},
number = {3},
pages = {505-523},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Lower semicontinuous functions with values in a continuous lattice},
url = {http://eudml.org/doc/247357},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Gool, Frans
TI - Lower semicontinuous functions with values in a continuous lattice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 3
SP - 505
EP - 523
AB - It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.
LA - eng
KW - continuous lattices; lower semicontinuous functions; potential theory; continuous lattice; semiuniform structure; order; Lawson topology
UR - http://eudml.org/doc/247357
ER -

References

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  1. Bliedtner J., Hansen W., Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, Berlin, 1986. Zbl0706.31001MR0850715
  2. Borwein J.M., Théra M., Sandwich Theorems for Semicontinuous Operators, preprint, 1990. 
  3. Bourbaki N., Topologie Générale, ch. IX, Hermann & Cie, Paris, 1948. MR0027138
  4. Constantinescu C., Cornea A., Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin, 1972. Zbl0248.31011MR0419799
  5. Fletcher P., Lindgren W.F., Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics 77, Marcel Dekker inc., New York, 1982. Zbl0583.54017MR0660063
  6. Gerritse G., Lattice-valued Semicontinuous Functions, Report 8532, Catholic University of Nijmegen, 1985. Zbl0872.54010
  7. Gierz G., Hofmann K., Keimel K., Lawson J., Mislove M., Scott D., A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980. Zbl0452.06001MR0614752
  8. van Gool F.A., Non-linear Potential Theory, Preprint 606, University of Utrecht, 1990. 
  9. Holwerda H., Closed Hypographs, Semicontinuity and the Topological Closed-graph Theorem: A unifying Approach, Report 8935, Catholic University of Nijmegen, 1989. 
  10. Katětov M., On real-valued functions in topological spaces, Fundamenta Mathematicae 38 (1951), 85-91 Correction in Fund. Math. 40 (1953), 203-205. (1953) MR0050264
  11. Nachbin L., Topology and Order, Van Nostrand, Princeton, 1965. Zbl0333.54002MR0219042
  12. Penot J.P., Théra M., Semi-continuous mappings in general topology, Arch. Math. 38 (1982), 158-166. (1982) 
  13. Tong H., Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19 (1952), 289-292. (1952) Zbl0046.16203MR0050265

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