2 research outputs found
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L-fuzzy compactness and related concepts
The compactness defined by Warner and McLean is extended to arbitrary L-fuzzy sets where L is a fuzzy lattice, i.e., a completely distributive lattice with an order reversing involution. It is shown that with our compactness we can build up a satisfactory theory. The different definitions of compactness in L-fuzzy topological spaces are stated and other characterizations of some of these notions are obtained. We also study their goodness and establish the inter-relations between the compactnesses which are good extensions.
Good definitions of L-fuzzy regularity and normality are proposed.
Following the lines of our compactness we suggest two definitions of L-fuzzy local compactness that are good extensions of the respective ordinary versions. A comparison between them is presented and some of their properties studied. A one point compactification is also obtained.
By introducing a new definition of a locally finite family of L-fuzzy sets and combining it with our definition of compactness, we propose an L-fuzzy paracompactness and study some of its properties.
Good definitions of L-fuzzy countable and sequential compactness and the Lindelof property are introduced and studied.
We also present, in L-fuzzy topological spaces, good extensions of S-closedness and RS-compactness. Some of their properties are examined.
Good L-fuzzy versions of almost compactness, near compactness and a strong compactness are put forward and studied. A comparison between these compactness related concepts is also presented
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Study of covering properties in fuzzy topology
This work is devoted to the study of covering properties both in L-fuzzy topological spaces and in smooth L-fuzzy topological spaces , that is the fuzzy spaces in Sostak's sense, where L is a fuzzy lattice . Based on the satisfactory theory of L-fuzzy compactness build up by Warner, McLean and Kudri, good definitions of feeble compactness and P-closedness are introduced and studied. A unification theory for good L-fuzzy covering axioms is provided.
Following the lines of L-fuzzy compactness, we suggest two kinds of L-fuzzy relative compactness as in general topology, study some of their properties and prove that these notions are good extensions of the corresponding ordinary versions.
We also present L-fuzzy versions of R-compactness , weak compactness and 0-rigidity and discuss some of their properties.
By introducing 'a-Scott continuous functions', a 'goodness of extension' criterion for smooth fuzzy topological properties is established. We propose a good definition of compactness, which we call 'smooth compactness' in smooth L-fuzzy topological spaces. Smooth compactness turns out to be an extension of L-fuzzy compactness to smooth L-fuzzy topological spaces. We study some properties of smooth compactness and obtain different characterizations. As an extension of the fuzzy Hausdorffness defined by Warner and McLean, 'smooth Hausdorffness' is introduced in smooth L-fuzzy topological spaces. Good definitions of smooth countable compactness, smooth Lindelofness and smooth local compactness are introduced and some of their properties studied