A characterization of the category Q-TOP

arXiv:1112.4315v1 [math.CT] 19 Dec 2011 A characterization of the category Q-TOP Sheo Kumar Singh∗ Department of Mathematics, Banaras Hindu University, Varanasi-221005, India Arun K. Srivastava† Department of Mathematics and Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi-221005, India 1 Introduction E.G. Manes in [2] gave a somewhat ‘axiomatic’ characterization (upto isomor- phism) of the category TOP of topological spaces, among a certain class of categories, satisfying some conditions, in which a Sierpinski space-like object, played a key role. Following that, the category [0, 1]-TOP of [0, 1]-topological spaces (known more commonly as ‘fuzzy’ topological spaces) was analogously characterized by Srivastava [4]. S.A. Solovyov [3] has recently introduced the no- tion of Q-topological spaces (Q being a fixed member of a variety of Ω-algebras). In this note, we give a characterization of the category Q-TOP of Q-topological spaces in which the so-called Q-Sierpinski space, introduced in [3], plays a key role. 2 The category Q-TOP We begin by recalling the (well-known) notions of Ω-algebras and their homo- morphisms. For details, see [1]. ∗sheomathbhu@gmail.com †arunksrivastava@gmail.com 1 Definition 2.1 ([3]) • Let Ω= (nλ)λ∈I be a class of cardinal numbers. An Ω-algebra is a pair (A, (ωA λ )λ∈I) consisting of a set A and a family of maps ωA λ : Anλ →A. Any B(̸= φ) ⊆A is called a subalgebra of (A, (ωA λ )λ∈I) if for any λ ∈I and any (bi)i∈nλ ∈Bnλ, ωA λ ((bi)i∈nλ) ∈B. Given S ⊆A, the intersection of all the subalgebras of (A, (ωA λ )λ∈I) containing S is clearly a subalgebra of (A, (ωA λ )λ∈I). We shall denote it as < S >. • Given (A, (ωA λ )λ∈I) and (B, (ωB λ )λ∈I), a map f : A →B is called an Ω-algebra homomorphism if for every λ ∈I the following diagram commutes: Anλ ωA λ  f nλ / Bnλ ωB λ  A f / B . Let Alg(Ω) denote the category of Ω-algebras and Ω-algebra homomor- phisms. • A variety of Ω-algebras is a full subcategory of Alg(Ω) which is closed under the formation of products 1, subalgebras and homomorphic images. Throughout, Q denotes a fixed member of a fixed variety of Ω-algebras. • Given a set X, a subset τ of QX is called a Q-topology on X if τ is a subalgebra of QX; in such a case, the pair (X, τ) is called a Q-topological space. • Given two Q-topological spaces (X, τ) and (Y, η), a Q-continuous func- tion from (X, τ) to (Y, η) is a function f : X →Y such that f ←(α) ∈ τ, ∀α ∈η, where f ←(α) = α ◦f. It is evident that all Q-topological spaces, together with Q-continuous maps, form a category, which we shall denote as Q-TOP. The following categorical concepts are from Manes [2] Definition 2.2 ([2]) 1. A category C of sets with structure is defined through the follow- ing descriptions of its objects (‘C-structured sets’) and morphisms (‘C- admissible maps’), satisfying the two axioms given below: A class C(X) of ‘C-structures’ is assigned with each set X and a ‘C- structured set’ is a pair (X, s), with s ∈C(X). 1The category Alg(Ω) is closed under products 2 A subset C(s, t) of the set of all functions from X to Y is assigned with each pair of C-structured sets (X, s) and (Y, t) and a ‘C-admissible map’ from (X, s) to (Y, t) is any f ∈C(s, t) (in which case we write “f : (X, s) →(Y, t)”). The axioms are: Axiom A1: If f : (X, s) →(Y, t) and g : (Y, t) →(Z, u) then also g ◦f : (X, s) →(Z, u). Axiom A2: Given a bijection f : X →Y and t ∈C(Y ), there exists a unique s ∈C(X) such that f : (X, s) →(Y, t) and f −1 : (Y, t) →(X, s). 2. Given a category C of structures (as defined above), a family F = {fj : (X, s) →(Yj, tj)|j ∈J} of C-admissible maps is said to be optimal if for each C-structured set (Z, u) and a function g : Z →X, g : (Z, u) →(X, s) ifffj ◦g : (Z, u) →(X, s), ∀j ∈J. Further, if for a family F = {fj : X →(Yj, tj)|j ∈J} of functions, where X is a set and each (Yj, tj) is a C-structured set, there exists s ∈C(X) such that the family {fj : (X, s) → (Yj, tj)|j ∈J} is optimal, then s is called an optimal lift of the family F. 3. An object S = (S, u) in a category C of sets with structure is called a Sierpinski object if for every C-object X = (X, s), the family of all C-admissible maps from X to S is optimal. Remark 2.1 It is easy to verify that, in Q-TOP, the optimal lift of a family F = {fj : X →(Yj, tj)|j ∈J} of functions, where X is a set and each (Yj, tj) is a Q-TOP-object, is precisely the smallest Q-topology on X making each fi Q-continuous. Both TOP and [0, 1]-TOP are categories of sets with structures and the usual two-point Sierpinski space and the fuzzy Sierpinski space (of [4]) are Sierpinski objects in these categories. We verify (on expected lines) that Q-TOP is also a category of sets with structures. For each set X, C(X) is the family of all Q-topologies on X and the C-admissible maps are just the Q-continuous maps. Given a bijection f : X →Y and t ∈C(Y ), there exists a unique s ∈C(X), namely s =< A >, with A = {q ◦f|q ∈t}, such that both f : (X, s)

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