Strongly Connectedness in Closure Space

A Čech closure space (X, u) is a set X with Čech closure operator u: P(X) → P(X) where P(X) is a power set of X, which satisfies u ф=ф, A ⊆uA for every A⊆X, u (A⋃B) = uA⋃uB, for all A, B ⊆ X. Many properties which hold in topological space hold in closure space as well. A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei‟s are nonempty disjoint closed subsets of X, then X≠ E1⋃E2⋃. We further extend the concept of strongly connectedness in closure space. The aim of this paper is to introduce and study the concept of strongly connectedness in closure space.


Introduction:
Čech closure space was introduced by Čech E. [1] in 1963. The modern notion of connectedness was proposed by Jorden (1893) and Schoenfliesz, and put on firm footing by Riesz [7] with the use of subspace topology. Many mathematicians such as EissaD.Habil, Khalid A. Elzenati [4], EissaD. Habil [5], Stadler B.M.R. and Stadler P.F. [6] have extended various concepts of strongly connectedness in topological space. In this paper, we introduce strongly connectedness in closure space and study some of their properties.
A ⊆uA , for every A⊆X, 3. u(A⋃B)=uA⋃uB , for all A, B⊆X. is called a Čech closure operator and the pair (X, u) is a Čech closure space.

Definition 2.2 [8]
:-A closure space (X, u) is connected if and only if there exists a continuous function from X to the discrete space {0, 1} is constant. A subset A in a closure space (X, u) is said to be connected if A with the subspace topology is a connected space.

Definition 2.3[3]:
A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei"s are nonempty disjoint closed subsets of X, then X≠ E1 E2 .............

Bulletin of Mathematical Sciences and Applications
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Strongly connectedness in closure space:
Definition 3.1: A connected closure space (X, u) is said to be strongly connected if and only if it cannot be expressed as a disjoint union of count ably many but more than one closed sets. In connected closure space (X, u), let E1 and E2 are two nonempty disjoint closed subsets of X then X≠E1 E2. In strongly connected closure space (X, u), Ei"s are nonempty disjoint closed subsets of X then X≠ E1 E2 .............

Definition 3.3:-
A connected closure space (X, u) is said to be strongly disconnected if and only if it can be expressed as a disjoint union of count ably many but more than one closed sets. Proof: Let (X, u) is a strongly connected closure space. Suppose f (X) is not strongly connected closure space then by definition it can be expressed as a disjoint union of count ably many but more than one closed sets. Since f is continuous and the inverse image of closed set is still closed, so that X can be expressed as a disjoint union of count ably many but more than one closed sets. Hence (X, u) is a strongly disconnected closure space which is a contradiction. Therefore f(X) is strongly connected closure space. is not strongly connected closure space so that X can be expressed as a disjoint union of count ably many but more than one closed sets i. e. X= Ei .Then define a function f: X→Z= {0, 1} by taking f(x) =i whenever x Ei. This f is continuous but not constant. Hence X is not z-connected closure space which is a contradiction. Therefore, X is strongly connected closure space. From the above theorem, strongly connected closure space is a special case of z-connectedness in closure space. Thus all the properties proved for z-connected closure space are applicable to strongly connected closure space. Theorem 3.7:-A strongly connected closure space is connected. But converse is not true as the following example shows. Theorem 3.9:-The union of any family of strongly connected subsets of strongly connected closure space with a common point is strongly connected closure space.
Proof:-Let (X, u) is a strongly connected closure space. Let each {Ei: i∈∧} is strongly connected subset of strongly connected closure space (X, u) and common point is y0.Let C= {⋃Ei: Ei⊆ X}, y0 ∈⋂Ei. For any continuous function f: C→ {0, 1}. Let ia: Ei→ C be the inclusion function. Each Ei is strongly connected, so that foia: Ei → {0, 1} is continuous and constant and ⋂ Ei ≠⌀, so there exists a y0 such that y0 ∈ ⋂ Ei , i. e. foia is constant and equal to f(y0 ). Therefore f is constant and ⋃Ei is strongly trongly connected Theorem 3.10:-Let A and B are subsets of a strongly connected closure space (X, u) such that A⊆B⊆A, where A is the closure of A. If A is strongly connected, then is strongly connected in closure space