Fuzzy T-neighbourhood spaces: Part 2—T-neighbourhood systems

doi:10.1016/S0165-0114(01)00156-7

Fuzzy Sets and Systems

Volume 127, Issue 3, 1 May 2002, Pages 265-280

https://doi.org/10.1016/S0165-0114(01)00156-7 Get rights and content

Abstract

We explore a notion of fuzzy $T$ -neighbourhood spaces, for any continuous triangular norm $T$ , and we present on this notion a unified treatment. Our theory, on one hand, generalizes the theory of Lowen (Fuzzy Sets and Systems 7 (1982) 65) from $T= Min$ to arbitrary $T$ , which has been the progenitor of this work, and on the other hand it is strongly related to the theory of $L$ -neighbourhoods of Höhle (Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht, 1999) (when $L$ is restricted to the lattice [0,1]). We study the $T$ -neighbourhood bases and systems of our spaces, their level topologies, as well as their relationship to the Lowen–Höhle fuzzy $T$ -uniformities (J. Math. Anal. Appl. 82 (1981) 370; Manuscripta Math. 38 (1982) 289) and to our fuzzy $T$ -proximities (introduced in Part 1). We show that the nearness concept underlying a fuzzy $T$ -neighbourhood space is a fuzzy relation of closeness between its ordinary subsets and ordinary points. In particular, our spaces are in canonical one-to-one correspondence with the ( $T$ -)probabilistic topological spaces of Frank (J. Math. Anal. Appl. 34 (1971) 67). We demonstrate that each full subcategory $T$ -FNS, of FTS, of fuzzy $T$ -neighbourhood spaces is a topological category. To do that, we characterize $|T$ - $FNS |$ within $|$ FTS $|$ , and we characterize continuity of functions within $T$ -FNS in terms of $T$ -neighbourhood bases.

References (28)

A.A. Allam et al.
On fuzzy neighbourhood spaces
Fuzzy Sets and Systems
(1991)
M.A. Amer et al.
Bounded linear transformations between probabilistic normed vector spaces
Fuzzy Sets and Systems
(1995)
A.B. El-Rayes et al.
Generalized possibility measures
Inform. Sci.
(1994)
M.J. Frank
Probabilistic topological spaces
J. Math. Anal. Appl.
(1971)
U. Höhle
Probabilistic uniformization of fuzzy topologies
Fuzzy Sets and Systems
(1978)
A. Kandil et al.
A level-topologies criterion for Lowen fuzzy uniformizability
Fuzzy Sets and Systems
(1994)
R. Lowen
Initial and final fuzzy topologies and the fuzzy Tychonoff theorem
J. Math. Anal. Appl.
(1977)
R. Lowen
A comparison of different compactness notions in fuzzy topological spaces
J. Math. Anal. Appl.
(1978)
R. Lowen
Fuzzy uniform spaces
J. Math. Anal. Appl.
(1981)
R. Lowen et al.
$FTS_{0}$ : The epireflective hull of the Sierpinski object in FTS
Fuzzy Sets and Systems
(1989)

A.S. Mashhour et al.

Semi-induced fuzzy topologies

Fuzzy Sets and Systems

(1989)

N.N. Morsi

A short note on fuzzy neighbourhood spaces

Fuzzy Sets and Systems

(1987)

N.N. Morsi

Nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces

Fuzzy Sets and Systems

(1989)

N.N. Morsi

Dual fuzzy neighbourhood spaces I

Fuzzy Sets and Systems

(1991)

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Citation Excerpt :
In this manuscript, we continue our study of T-syntopogenous spaces. We show how the T-syntopogenous spaces agree well with the fuzzy T-neighbourhood spaces [4], fuzzy T-uniform spaces [9], fuzzy T-proximity spaces [3] and syntopogenous spaces [2]. More precisely, it is shown that there is a one-to-one correspondence between the fuzzy T-neighbourhood spaces and the so-called perfect T-syntopogenous spaces.
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We end by developing complete syntax (with fewer axioms and inference rules) for the smaller semantical domain of all adjointness algebras whose implications satisfy the exchange principle.

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Fuzzy T-neighbourhood spaces: Part 2—T-neighbourhood systems

Abstract

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Inform. Sci.

J. Math. Anal. Appl.

Fuzzy Sets and Systems

Fuzzy Sets and Systems

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy $T$ -neighbourhood spaces: Part 2— $T$ -neighbourhood systems