Foundations of the theory of (L,M)-fuzzy topological spaces

Introduction and motivation To explain the motivation of our work we give a short glimpse into the history of Fuzzy Topology called more recently Lattice-Valued Topology or Many Valued Topology. In 1968, C. L. Chang [1] introduced the notion of a fuzzy topology on a set X as a subset τ ⊆ [0, 1] satisfying the natural counterparts of the axioms of topology: (1) 0X , 1X ∈ τ ; (2) U, V ∈ τ ⇒ U ∧ V ∈ τ ; (3) U ⊆ τ ⇒ ∨ U ∈τ. Five years later, J. A. Goguen [2] replaced the interval [0, 1] with an arbitrary complete infinitely distributive lattice L thus obtaining the concept of an L-fuzzy topology or just an L-topology. In 1980, U. Höhle [3] came to the concept of an L-fuzzy topology being an L-subset T of the powerset P(X) ≈ 2 , that is a map T : P(X) → L such that: (1) T (∅) = T (X) = 1, (2) T (U ∩ V ) ≥ T (U) ∧ T (V ) for any U, V ∈ P(X), and (3) T ( ∨ U) ≥ ∧ T (U) for all U ⊆ P(X). The latter concept is now called a fuzzifying topology, after the 1991 paper by Mingsheng Ying [4]. Finally, in 1985, the authors of this abstract independently introduced the concept of an L-fuzzy topology X as a map T : L → L such that: (1) T (0X) = T (1X) = 1; (2) T (U ∧ V ) ≥ T (U) ∧ T (V ) ∀U, V ∈ L ; (3) T ( ∨ U) ≥ ∧ T (U) ∀U ⊆ L ; see [5], [10], [11]. For historical reason we note that in [10] the case L = [0, 1] was considered and developed, while [5] merely introduces (without further developing) fuzzy topologies of the form T : L → M with L and M being complete lattices in a variable-basis setting á la S. E. Rodabaugh [8] (see next section). Note also that some authors (J.A. Goguen [2] was the first) consider topological-type structures in the context of L-fuzzy sets in case when L is endowed with an additional binary operation ∗ which allows to introduce residuation in L. However, we will not scare this direction since our interest here concerns mainly the role of the lattice properties of L in the research of topological-type structures in the context of fuzzy sets.