Fuzzy topology

lattice-valued topology, point-set lattice-theoretic topology, poslat topology

A branch of mathematics encompassing any sort of topology using lattice-valued subsets. The following description is based on the standardization of this discipline undertaken in [a9], especially [a10], [a11]; much additional information is given in the references below.

Let be a set and any complete quasi-monoidal lattice (a cqml; i.e., is a complete lattice with bottom element and top element , and the tensor product is isotone in both arguments with ). Examples of complete quasi-monoidal lattices are:

complete lattices with (binary meet);

with any of the -norms , , or ; or

with any of , , etc.

-subsets of comprise the -powerset , a complete quasi-monoidal lattice via the lifting of the structure of . A subfamily is an -topology on , and is an -topological space, if is closed under and arbitrary and contains the constant mapping . A function is an -fuzzy topology on , and is an -fuzzy topological space, if satisfies (reading as "for each" ):

1) set , ,

2) two-element set , ,

3) .

The member of is interpreted as the "degree of openness" of .

Important examples of -topological and -fuzzy topological spaces can be found in [a5], Chap. 11; [a6], Kubiak's paper; [a8]; [a9], Chaps. 6, 8, 10; [a9], Chap. 7, Sect. 2.15–2.16; [a10], Sect. 7; [a11], Sect. 7.

For a complete quasi-monoidal lattice and function , one defines the powerset operators (the image operator) and (the pre-image operator) by

It is well-known that and that these operators generalize the traditional operators and . Given -topological spaces and , a mapping is -continuous from to if ; and given -fuzzy topological spaces and , a mapping is -fuzzy continuous from to if on . The category - comprises -topological spaces, -continuous mappings, and the composition and identities from the category (cf. also Sets, category of); and the category - comprises -fuzzy topological spaces, -fuzzy continuous mappings, and the composition and identities from the category . It is a theorem that for all complete quasi-monoidal lattices , the categories - and - are topological categories over , in the sense of [a3] and [a11], Sect. 1, and hence topological constructs.

The above briefly describes "fixed-basis topology" — topology where the complete quasi-monoidal lattice , viewed as the lattice-theoretic base of powersets and spaces or , is fixed relative to the spaces and mappings of the category - or -. "Variable-basis topology" permits the base to change within a category, so that each space has its own lattice-theoretic base.

To outline variable-basis topology, note that all complete quasi-monoidal lattices form a category, , in which morphisms are mappings preserving , arbitrary , and ; and also note that and embed into . One then considers , with objects the same as those of , now called localic quasi-monoidal lattices, but morphisms reversed from those of ; and one notes that and embed into .

Now, let . The category - for variable-basis topology and the category - for variable-basis fuzzy topology are both "concrete" categories over as a "ground" or "base" category. For a morphism , the pre-image operator is defined by . An image operator is also available which, if preserves arbitrary , satisfies ; and if , these operators reduce to their fixed-basis counterparts.

Data for the category - include:

objects are topological spaces (cf. also Topological space), where and ;

morphisms are continuous mappings

(cf. also Continuous function), where is in and .

Data for the category - include:

objects are fuzzy topological spaces , where and ;

morphisms are fuzzy continuous mappings , where is in and on .

In both categories, compositions and identities are those of .

It is a theorem that for all , - and - are topological over the ground in the sense of [a3] and [a11], Sect. 1. Further, these frameworks unify all the fixed-basis categories for topology given above and hence unify all important examples (referenced above) over different lattice-theoretic bases (e.g. two fuzzy real lines and ). Moreover, all purely lattice-theoretic or point-free approaches to topology — locales, topological molecular lattices, uniform lattices, etc. (see [a1], [a2], [a8]) — categorically embed into - or - (for appropriate ) as subcategories of singleton spaces; e.g. embeds as singleton spaces into -. Thus, the variable-basis approach categorically unifies topology and fuzzy topology as a discipline.

References

[a1]	P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1982)
[a2]	G.-J. Wang, "Theory of L-fuzzy topological spaces" , Shanxi Normal Univ. Publ. House (1988) (In Chinese)
[a3]	J. Adamek, H. Herrlich, G.E. Strecker, "Abstract and concrete categories" , Wiley (1990)
[a4]	U. (ed.) Höhle, "Mathematical aspects of fuzzy set theory" Fuzzy Sets and Syst. , 40 : 2 (1991) (Special Memorial Volume–Second Issue)
[a5]	"Applications of category theory to fuzzy subsets" S.E. Rodabaugh (ed.) E.P. Klement (ed.) U. Höhle (ed.) , Kluwer Acad. Publ. (1992)
[a6]	U. Höhle, S.E. Rodabaugh, A. (eds.) Šostak, "Special issue on fuzzy topology" Fuzzy Sets and Syst. , 73 : 1 (1995)
[a7]	W. (ed.) Kotzé, "Special issue" Quaestiones Math. , 20 : 3 (1997)
[a8]	Liu Ying–Ming, Luo Mao–Kang, "Fuzzy topology" , World Sci. (1997)
[a9]	"Mathematics of fuzzy sets: Logic, topology, and measure theory" U. Höhle (ed.) S.E. Rodabaugh (ed.) , The Handbooks of Fuzzy Sets , 3 , Kluwer Acad. Publ. (1999)
[a10]	U. Höhle, A. Šostak, "Axiomatic foundations of fixed-basis fuzzy topology" U. Höhle (ed.) S.E. Rodabaugh (ed.) , Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory , The Handbooks of Fuzzy Sets , 3 , Kluwer Acad. Publ. (1999) pp. 123–272
[a11]	S.E. Rodabaugh, "Categorical foundations of variable-basis fuzzy topology" U. Höhle (ed.) S.E. Rodabaugh (ed.) , Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory , The Handbooks of Fuzzy Sets , 3 , Kluwer Acad. Publ. (1999) pp. 273–388