Compact \({\mathcal G}\)-fuzzy topological spaces
type: Item
instance of: scholarly article
title: Compact \({\mathcal G}\)-fuzzy topological spaces
zbMATH Open document ID: 0572.54005
DOI: 10.1016/0165-0114(84)90025-3
author: Ulrich Höhle
published in: Fuzzy Sets and Systems
publication date: 1984-01-01T00:00:00Z
review text: Do convergence and compactness properties of L-fuzzy topological spaces require the complete distributivity of the underlying lattice L? The author examines this question when L is a complete Boolean algebra because, if the answer were positive, the theory would be expected to have no significant application to probability theory. In fact, the paper provides a negative answer to the question for the special case considered. The author develops a convergence theory which does not require complete distributivity by replacing the prime prefilters of \textit{R. Lowen} [Gen. Topol. Appl. 10, 147-160 (1979; Zbl 0409.54008)] with a new concept of ultrafilters. As a result he can define a notion of compactness (called probabilistic compactness in the paper) which has properties parallel to ordinary compactness. For example, it can be characterized by a combination of completeness and pre-compactness. He also shows that, for any ordinary compact space X, the space of almost- everywhere defined X-valued random variables is probabilistic compact in a natural way.
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 3913221
zbMATH Keywords: projective limits
zbMATH Keywords: unique compatible fuzzy uniformity
zbMATH Keywords: 1-ultrafilter
zbMATH Keywords: probabilistic (fuzzy) compactness
zbMATH Keywords: regularity
zbMATH Keywords: probabilistic precompactness
zbMATH Keywords: probabilistic completeness
zbMATH Keywords: L-fuzzy topological spaces
zbMATH Keywords: complete distributivity of the underlying lattice
MaRDI profile type: Publication
full work available at URL: 0165-0114(84)90025-3
OpenAlex ID: W2000330860
cites work: Q5809017
cites work: Q5331549
cites work: Q3256687
cites work: L-fuzzy sets
cites work: Probabilistische Topologien
cites work: Probabilistisch kompakte L-unscharfe Mengen
cites work: Probabilistic uniformization of fuzzy topologies
cites work: Representation theorems for L-fuzzy quantities
cites work: Probabilistic topologies induced by \(L\)-fuzzy uniformities
cites work: Normality in fuzzy topological spaces
cites work: Uniformities on fuzzy topological spaces
cites work: Products of fuzzy topological spaces
cites work: Q5531649
cites work: Fuzzy topological spaces and fuzzy compactness
cites work: Initial and final fuzzy topologies and the fuzzy Tychonoff theorem
cites work: Convergence in fuzzy topological spaces
cites work: The Hausdorff separation axiom for fuzzy topological spaces
cites work: Q5585437
Recommended article: Q5455029
Recommended article: $\alpha$-compact fuzzy topological spaces
Recommended article: \(\theta\)-compact fuzzy topological spaces
Recommended article: On the topological structure of spaces of fuzzy compacta
Recommended article: Q3697783
Recommended article: Q4037041
Recommended article: Q3312883
Recommended article: Compactness in \(L\)-fuzzy topological spaces
Recommended article: Q4898322
Recommended article: Q5287240
description: scientific article; zbMATH DE number 3913221