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 |
Fuzzy topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fuzzy_topology&oldid=15614