18-09-2014, 09:51 AM
Pairwise Almost δ Semicontinuous Functions
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Abstract
The aim of this paper is to introduce the notion of and study their basic properties in bitopological spaces.
Introduction
The concept of bitopological space was introduced by J. C. Kelly[7]. By using various forms of open sets many authors introduced and studied various types of continuity. In 1980, Noiri[12] studied the notion of δ continuous function. Recently Erdal Ekici[6] was introduced the notion of almost δ semicontinuous function. In this paper is to introduce the notion of function and we obtain its some characterizations.
Preliminaries
Throughout this paper, and always mean bitopological spaces. For a function f : denotes a function f of a space into a space . For A subset of , τj - Cl ( A ) ( resp. τi - iInt ( A ) ) denotes the closure ( resp. interior ) of A, where i ≠ j ; i, j = 1, 2. However
τj - Cl ( A ) and τi - iInt ( A ) are briefly denoted by jCl ( A ) and iInt ( A ) respectively
Definition 2.1
A subset A of a bitopological space is said to be
( i )
( ii )
(iii)
Definition 2.3 :
A function f : is said to be
(i) semi open [2] if for each τi¬ - open set U of , f ( U ) is ( i, j ) semi open in . where i ≠ j; i, j = 1, 2
(ii) preopen [8] if for each τi¬ - open set U of ,
f ( U ) is preopen in . where i ≠ j; i, j = 1, 2
Definition 2.4 :
A subset A of a bitopological space is said to be δ semi open if A jCl [ δ Int( A ) ]. where i ≠ j; i, j = 1, 2.
The complement of δ semi open is called a δ semi closed.
The union ( resp. intersection ) of all δ semi open ( resp. δ semi closed )sets each contained in ( resp. containing ) a set A in a bitopological space is called the δ semiinterior ( resp. δ semiclosure ) of A and it is denoted by δ sint ( A ) ( resp. δ scl (A ) ).
The family of all δ semi open ( resp. regular open, preopen , open , semi open, δ open ) sets of a space will be denoted by δ SO ( resp. RO , PO , O , SO , δO ).
The family of all δ semi closed ( resp. regular closed , δ closed ) sets in a space X is denoted by δ SC ( resp. RC , δC ).
The family of all δ semi open ( resp. regular open , δ open ) sets containing a point x X will be denoted by δ SO ( X , x ) ( resp. RO ( X , x) , δ O ( X , x ) ).
Definition 3.1 :
A function f : is said to be
(i) semicontinuous if f ⁻ ¹ ( V ) - SO for every
σi - open subset V of . where i ≠ j; i, j = 1, 2.
(ii) almost semicontinuous if f -1 ( V ) - SO for every σi - open set V - RO . where i ≠ j; i, j = 1, 2.
(iii) δ semicontinuous if f ⁻ 1 ( V ) is δ semi open in for every
σi - open set V of . where i ≠ j; i, j = 1, 2.
(iv) δ continuous if f -1 ( V ) is δ open in for every σi - open set V RO . where i ≠ j; i, j = 1, 2.
Theorem 3.7: For a function f : the following are equivalent:
[ 1 ] f is almost δ semicontinuous.
[ 2 ] f [ δ scl ( A ) ] δ cl [ f ( A ) ] for every subsets A of .
[ 3 ] δ scl [f ⁻¹( B ) ] f ⁻¹[ - δ cl ( B ) ] for every subset B of .
[ 4 ] f ⁻¹ ( F ) δ SC for every σj δ closed set F of .
[ 5 ] f ⁻¹ ( V ) δ SO for every σi δ open set V of .
Proof: ( 1 ) ( 2 )
Let V be any σi open set of . It follows from[ 2,Theorem 2.4] that
jCl ( V ) is σj regular closed in .
Since f is almost δ semicontinuous, by theorem 3.4,
f ⁻¹ [ jCl ( V ) ] is δ semiclosed in .
Therefore , ( i, j ) δ scl [ f ⁻¹ ( V ) ] f ⁻¹ [ jCl ( V ) ].
( 2 ) ( 3 ) This is obvious. since ( i, j ) SO ( i, j ) O .
( 3 ) ( 1 ) Let V be any σj regular closed set of .
Then V is σi semiopen in .
Hence ( i, j ) δ scl [ f ⁻¹ ( V ) ] f ⁻¹ [ jCl ( V ) ] = f ⁻¹ ( V ).
This shows that f ⁻¹ ( V ) is δ semiclosed. Therefore, by theorem3.4,
f is almost δ semicontinuous.