T-Semi Connected Spaces

In this paper, we i ntroduce a new co ncept, namely T-semi connected space, where T is an operator associated with the to pological T defined on a non-empty set X. Several properties of this concept are proved.


1-Introduction:
In [4], the concept of semi connected set is given.In this paper, we introduce the concept of T-semi connected set and show that it generalizes the concept of semi connected set when the operator T is the identity operator.
Throughout this paper, we use the following notations: cl(A) denotes the usual closure and int(A) denotes the interior of a set A.

2-Basic Definitions and Results:
In this section, we recall and introduce the basic definitions needed in this work.

Definition (2.1):
Let (X, τ) be a topological space and let T be an operator associated with τ, (X, τ, T) is called an operator topological space, [2].
Let A ⊆ X, we say that A is Tsemi open if there exists an open set U ∈ τ, such that: The Before we state the next proposition, we recall the following definition:

Definition (2.3), [2]:
Let (X, τ, T) be an operator topological space.We say that T is a monotone operator if for every pair of open sets U and V, such that U ⊂ V, we have that T(U) ⊂ T(V).
The closure operator is a monotone operator.

Proposition (2.4):
Let (X, τ, T) be an operator topological space, where T is a monotone operator

Remark (2.5):
If T is the closure operator, then A is T-semiopen set if and only if A ⊆ cl(int(A)), and this agrees with the equivalence given by Levin in [1].

Proposition (2.6):
Let (X, τ, T) be an operator topological space, where T is the closure operator and let S ⊆ X.Then S is T-semi closed if and only if there exists a closed subset F of X, such that int(F) ⊆ S ⊆ F.

Proof:
Suppose that: Conversely, if S is Tsemiclosed Then S c is T-semiopen Therefore, there exists an open set U, such that:

Remark (2.7): [3]
Let (X, τ, T) be an operator topological space, where T is the closure operator.Then a subset S of X is T-semiclosed if and only if int(T(S)) ⊆ S.

Definition (2.8):[3]
Let (X, τ, T) be an operator topological space, where T is a monotone operator and let S ⊆ X , then the union of all T-semi open sets contained in the set S is T-semi open, and it is denoted by T-sInt(S).
We show this as follows :

Remark (2.9):[3]
If T is a monotone operator, the intersection of all T-semi closed sets of X containing the set S is T-semi closed.It is called the T-semi closure of S and is denoted by T-scl(S).
So, S is T-semi closed if and only if T-scl(S) = S.
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Main Results:
In this section, several properties and characterizations of Tsemi connected spaces are given.
, that is, if W cannot be expressed as the union of two T-semi separated sets.(iv) (X, τ, T) is said to be T-semi connected if and only if X is Tsemi connected.

Remark (3.2):
Let (X, τ, T) be an operator topological space, where T is the identity operator (In this case we write (X, τ) instead of (X, τ, T)) let A ⊆ X , B ⊆ X.The definition of Tsemi separated sets agrees with the definition of separated sets in the usual sense cl(A) ∩ B =A ∩ cl(B) = ∅ and therefore the definition of T-semi connected set generalizes the definition of connected set.

Example:
Let (X, τ, T) be an operator topological space consider X= ( IR , t u ) and where T is the closure operator).

Theorem (3.3):
Let (X, τ, T) be an operator topological space.Then X is T-semi connected if and only if the only subsets of X that are both T-semi open and T-semi closed in X are the empty set and X itself.

Proof:
Let A ⊆ X be a non-empty proper subset of X, which is both Tsemi open and T-semi closed in X, then the sets U = A and V = A c constitute a T-semi separation of X.So X will be T-semi disconnected.
Conversely, if U and V form Tsemi separation of X and X = U ∪ V, then U is a non-empty and different from X, since: complement of a T-semi open set is called a T-semi-closed set.Remarks (2.2): (i) If T is the closure operator (T(A) = cl(A)), the above definition agrees with the definition of semi-open set which is given by Levin in [1].(ii) If T is the identity operator (T(A) = A) the definition of T-semiopen set agrees with the definition of open set.(iii) For any operator T, each open set is T-semi-open set.Example Let (X, τ, T) be an operator topological space, where T is the closure operator and Tsemi open set will be the usual semiopen set ( the usual semi open set G ⊆ W ⊆ − G = CL (G) and Tsemi open set G ⊆ W ⊆ T (G) ) Consider X = ( IR, tu) and A= [ 0,1) is the Tsemi open set ( T is the 5957 closure operator ) but A is not open set.
We obtain that both sets U and V are T-semi open and T-semi closed.< Theorem (3.4):Let (X, τ, T) be an operator topological space, if A ⊆ X is a Tsemi connected and A ⊆ C ∪ D, where C and D are T-semi separated sets, then either A ⊆ C or A ⊆ D. Proof:A = A ∩ (C ∪ D) = (A ∩ C) ∪ (A ∩ D) Since C and D are T-semi separated sets C ∩ (T−scl(D)) = ∅ so (A ∩ C) ∩ (T−scl(A)) ∩ (T−scl(D)) ⊆ C ∩ (T−scl(D)) = ∅ So if both A ∩ C ≠ ∅ and A ∩ D ≠ ∅, then A is T-semi disconnected This shows that either A ∩ C = ∅, or A ∩ D = ∅ So A ⊆ C, or A ⊆ D. <PDF created with pdfFactory Pro trial version www.pdffactory.com