Connectedness in Isotonic Spaces

Turk J Math 30 (2006) , 247 – 262. c TÜBİTAK Connectedness in Isotonic Spaces Eissa D. Habil, Khalid A. Elzenati Abstract An isotonic space (X, cl) is a set X with isotonic operator cl : P(X) → P(X) which satisfies cl(∅) = ∅ and cl(A) ⊆ cl(B) whenever A ⊆ B ⊆ X. Many properties which hold in topological spaces hold in isotonic spaces as well. The notion of connectedness that is familiar from topological spaces generalizes to isotonic spaces. We further extend the notions of Z-connectedness and strong connectedness to isotonic spaces, and we indicate the intimate relationship between these notions. Key Words: generalized closure spaces, isotonic spaces, neighborhood spaces, connectedness, Z-connectedness, strong connectedness. 1. Generalized Closure Spaces Closure spaces and (more generally) isotonic spaces have already been studied by Hausdorff [13], Day [3], Hammer [11, 12], Gnilka [6, 7, 8] and Stadler [15, 16]. In this paper we explore some meaningful topological concepts that can be defined for isotonic spaces, especially the connectedness. Let X be a set, P(X) its power set and cl : P(X) → P(X) be an arbitrary set-valued set-function, called a closure function. We call cl(A), A ⊆ X, the closure of A, and we call the pair (X, cl) a generalized closure space. Consider the following axioms of the closure function for all A, B, Aλ ∈ P(X): K0) cl(∅) = ∅. Kl) A ⊆ B implies cl(A) ⊆ cl(B) (isotonic). AMS Mathematics Subject Classification: 54A05 247 HABIL, ELZENATI K2) A ⊆ cl(A) (expanding). K3) cl(A ∪ B) ⊆ cl(A) ∪ cl(B) K4) cl(cl(A)) = cl(A) K5) S (sub-additive). (idempotent). S λ∈Λ cl(Aλ ) = cl( λ∈Λ (Aλ )) (additive). The dual of a closure function is the interior function int : P(X) → P(X) which is defined by int(A) := X \ cl(X \ A). (1) Given the interior function int : P(X) → P(X), the closure function is recovered by cl(A) := X \ (int(X \ A)) for all A ∈ P(X). (2) A set A ∈ P(X) is closed in the generalized closure space (X, cl) if cl(A)=A holds. It is open if its complement X \ A is closed or equivalently A = int(A). It should be noted that the open and closed sets will not play a central role in our discussion. From now on, (for short) the word space will mean a generalized closure space. Definition 1.1 [15] Let cl and int be a closure and its dual interior function on X. Then the neighborhood function N : X → P(P(X)) assigns to each x ∈ X the collection N(x) := { N ∈ P(X) | x ∈ int(N )} (3) of its neighborhoods. A set V is a neighborhood of A, i.e. V ∈ N(A), if V ∈ N(x) ∀x ∈ A. The proof of the next lemma follows immediately from the definitions. Lemma 1.1 [15, 16] For any space (X, cl), V ∈ N(A) if and only if A ⊆ int(V ). The next theorem illustrates the intimate relationship between closures of sets and neighborhoods of points. Theorem 1.1 [15, 16] Let N be the neighborhood function defined in equ.(3). Then x ∈ cl(A) if and only if X \ A ∈ / N(x). It should be noted that there are equivalent properties for (Ki), i = 0, 1, ..., 5, which can be expressed in terms of interior or neighborhood functions (see [15, 16, 10]). 248 HABIL, ELZENATI 2. Isotonic Spaces Definition 2.1 [15, 16] An isotonic space is a pair (X, cl), where X is a set and cl : P(X) → P(X) satisfies the axioms (K0) and (K1). An isotonic space (X, cl) that satisfies (K2) is called a neighborhood space. A closure space is a neighborhood space that satisfies (K4). A topological space is a closure space that satisfies (K3). Lemma 2.1 [12, Lemma10] The following conditions are equivalent for an arbitrary closure function cl : P(X) → P(X): (K1) A ⊆ B ⊆ X implies cl(A) ⊆ cl(B). (K1I ) cl(A) ∪ cl(B) ⊆ cl(A ∪ B) for all A, B ∈ P(X). (K1II ) cl(A ∩ B) ⊆ cl(A) ∩ cl(B) for all A, B ∈ P(X). It is easy to derive equivalent conditions for the associated interior function by repeated application of int(A) = X \ cl(X \ A), as the following lemma shows. Lemma 2.2 [15] The following conditions are equivalent for an arbitrary interior function: int : P(X) → P(X): (K1III ) A ⊆ B ⊆ X implies int(A) ⊆ int(B). (K1IV ) int(A) ∪ int(B) ⊆ int(A ∪ B) for all A, B ∈ P(X). (K1V ) int(A ∩ B) ⊆ int(A) ∩ int(B) for all A, B ∈ P(X). An isotonic space can be described by means of interior and neighborhood functions, as the following two lemmas show. Their proofs follow immediately from the definitions and, therefore, are omitted. Lemma 2.3 A space (X, cl) is isotonic if and only if the interior function int : P(X) → P(X) satisfies: I0) int(X) = X; I1) int(A) ⊆ int(B) ∀A ⊆ B ⊆ X. Lemma 2.4 A space (X, cl) is isotonic if and only if the neighborhood function N : X → P(P(X)) satisfies: 249 HABIL, ELZENATI N0) X ∈ N(x) ∀x ∈ X; N1) N ∈ N(x), N ⊆ N1 implies N1 ∈ N(x). The next theorem shows that (N1)(or equivalently (K1)) is a necessary and sufficient condition for defining the closure function in terms of neighborhoods. Theorem 2.1 [3, Theorem 3.1, Corollary 3.2] Let (X, cl) be a space and c(A) := {x ∈ X | ∀N ∈ N(x) : A ∩ N 6= ∅} for all A ⊆ X. Then (i) c(A) ⊆ cl(A). (ii) c : P(X) → P(X) is isotonic (i.e., satisfies (K1)). (iii) c(A) = cl(A) if and only if cl is isotonic. The following brief study of the lower separation axiom T1 for isotonic and neighborhood spaces is needed for the study of connectedness in section 5. 0 Definition 2.2 [15, 16] A space (X, cl) is a T1 -space if ∀x, y ∈ X, x 6= y, ∃N ∈ N(x) 00 00 0 /N , y∈ /N . and N ∈ N(y) such that x ∈ Proposition 2.1 [15] An isotonic space (X, cl) is a T1 -isotonic space if and only if cl({x}) ⊆ {x} ∀x ∈ X. In general topology the definition of T1 -topological spaces has several equivalent forms ([5, Theorem V-1.2], [17, Theorem 13.4]), which can be generalized to neighborhood spaces, as in the next theorem. Theorem 2.2 Let (X, cl) be a neighborhood space. Then the following statements are equivalent: a) (X, cl) is T1 . b) Each one-point set in X is closed. c) Each subset of X is the union of closed sets contained in it. Proof. Let (X, cl) be a neighborhood space. 250 HABIL, ELZENATI (a ⇒ b): If (X, cl) is T1 , then, by Proposition 2.1, cl({x}) ⊆ {x} for all x ∈ X. Since, by (K − 2), {x} ⊆ cl({x}), we have cl({x}) = {x} ∀x ∈ X. Therefore, ∀x ∈ X, {x} is a closed set. (b ⇒ c): It is trivial. (c ⇒ a): If each subset of X is the union of closed subsets contained in it, then it is 2 trivial that {x} = cl{x} ∀x ∈ X. Therefore, by Proposition 2.1, (X, cl) is T1 . 3. Continuous Functions The purpose of this section is to define continuous functions on a space (X, cl) with arbitrary closure function, and establish their elementary properties. Definition 3.1 [15, 16] Let (X, cl) and (Y, cl) be two spaces. A function f : X → Y is continuous if cl(f −1 (B)) ⊆ f −1 (cl(B)) ∀B ∈ P(Y ). Proposition 3.1 [16] Let (X, cl) and (Y, cl) be two spaces and following statements are equivalent: f : X → Y . Then the (1) f is continuous. (2) f −1 (intY (B)) ⊆ intX (f −1 (B)) (3) B ∈ NY (f(x)) ∀B ∈ P(Y ). implies f −1 (B) ∈ NX (x) ∀B ∈ P(Y ). Definition 3.2 (...truncated)