Function- -Chain in Bitopological Spaces
Vijeta Iyer1, K.Meena2 , S.R.Saratha3
Assistant Professor, Department of Science and Humanities (Mathematics), Kumaraguru College of Technology, Coimbatore, India1,2,3
ABSTRACT:In this paper, the concept of − that was introduced using continuous functions in [1] for topological spaces has been extended to bitopological spaces. Function – – chainability has been defined in terms of real valued continuous functions on the space. Several well known results prove perquisites to results established in the paper.
KEYWORDS: − − chain, uniform − − chainability, uniform chainability,
− − chain preserving map, chain preserving map, strongly − − chain preserving map, strong chain preserving map.
Subject Classification: AMS (2000) : 54 A 99.
I.INTRODUCTION
As long back as in 1883, Cantor defined connectedness in metric spaces through -chains which have been studied extensively by many mathematicians. These -chains were defined between two given points and comprised points of the metric space under consideration. -Chainability characterizes connected sets among compact sets in the setting of metric spaces. In the same setting Beer [1] has characterized compact sets among the connected sets. Different definitions of chains in metric spaces have been provided by various Mathematicians. A simple chainjoining two points p, qof the metric space Xis a finite sequence A1, A2,……….,Am of subsets of X such that A1 (and only A1) contains p,
Am (and only Am) contains q and Ai∩Aj=øif and only if |i-j|>1. However, Piotr Minc [10] has called such a chain by
the name weak chain, whereas Newmann [9]has called it a chain of sets. S. A. Naimpally and C.M. Pareek [7] have generalized the concept of -chainability in metric spaces to topologicalspaces by the use of open covers and have obtained a characterization of connectedness in terms of open cover. D.P. Bellamy [2] defined the concept of connectedness by uniformly short paths. Kiran Shrivastava andGeeta Agrawal [11] have defined chainable sets using chains between points.Vijeta Iyer and Kiran Shrivastava [12] have extended these concepts to topological spaces through continuous functions. Function – – chainability has been defined in terms of real valued continuous functions on the space. In this article, the same concept [12] has been extended to bitopological spaces.
Throughout this paper [0,∞) has usual metric topology and is positive real number unless mentioned otherwise. Also is the intersection of two topologies and on X. If ( , , ) is a bitopological space and A is any subset of X then ( , , ) is a bitopological subspace of ( , , ) where ( , ) is a topological space and are relative topologies on the A.
II. PRELIMINARIES
Definition 2.2: Let ( , ) be a topological space and let there exist a non-constant continuous function : →[0,∞)
such that is function – – – chainable for every > 0. Then is said to be – − ℎ .
Definition 2.3: A topological space ( , ) is said to be − − − ℎ if for every > 0
there exists a positive integer ( ) and a non-constant continuous function : →[0,∞) such that for every pair of elements , of there is a sequence = , , … , = in with ≤ ( ) and | ( )− ( )| <
; 1≤ ≤ .
III. FUNCTION- − –CHAINS
Definition 3.1: A bitopological space ( , , ) is said to be function– − − −chainable if for any > 0 there exists a non-constant continuous function : ( , ∩ )→[0,∞) such that for every pair of points , of , there exists a sequence = , , … , = of points in with | ( )− ( )| < ; 1≤ ≤ .
Definition 3.2: Let ( , , ) be a bitopological space and let there exist a non-constant continuous function
: ( , )→[0,∞), ℎ = ∩ such that is function – − – – chainable for every > 0. Then is
said to be function−bi− −chainable.
Definition 3.3: Let( , , ) be a bitopological space, τ=τ ∩ τ and ⊂ . Then ( , , ) is said to be
function− − − −chainable if ( , ) is function– − – –chainable subspace of ( , ).
Definition 3.4: Let ( , , ) be a bitopological space, τ=τ ∩ τ and ⊂ . Let thereexist a non-constant continuous function ∶( , )→[0,∞) such that ( , ) is function– bi− – – chainable for every > 0. Then
( , , ) is said to be −bi− − ℎ .
Definition 3.5: A bitopological space ( , , ) is said to be uniformlyfunction−bi− − −chainable if for
> 0 there exists a positive integer ( ) and a non-constant continuous function : →[0,∞) such that for every
pair of elements , of there is a sequence = , , … , = in with ≤ ( ) and| ( )− ( )| <
; 1≤ ≤ .
Definition 3.6: Let( , , ) be a bitopological space and let there exist a non-constant continuous function : →
[0,∞) such that is uniformly function – − – – chainable for every > 0. Then is said to be
uniformlyfunction−bi− −chainable.
Theorem 3.1: Let ( , , ) be a bitopological space and let be the intersection of and . Then if : ( , )→
[0,∞) is continuous onto function then ( , , ) is function –bi – – chainable.
Proof: Let , ∈ and > 0. Let be the least positive integer greater than (| ( )− ( ′)|)⁄ .Without loss of generality, let ( ) > ( ).
Choose
= ( ), = +| ( )− ( )|, = +2(| ( )− ( )|),⋯ , = ( ) [0,∞).
Also | − | < ; 1≤ ≤ .
Then there exists a sequence of points , , , …, , in such that | ( )− ( )| = | − | < . or
( , , ) is function – − – chainable.
Theorem 3.2: The relation of function – − – ε – chainability in a bitopological space is an equivalence relation.
Theorem 3.3: Let ( , , ) be a bitopological space and ⊂ . If for every ε> 0 there exists a continuous function
∶( , )→[0,∞), where be the intersection of and such that is function – − – – chainable, then
( , , ) is function – − – – chainable.
Proof: Let , ∈ . As ⊂ ( ) , ( ), ( )∈ ( ) or there exist , ∈ such that | ( )− ( )| < and
| ( )− ( )| < . Hence there exist a sequence of elements = , , … , = ′ in such that
| ( )− ( )| = | ( )− ( )| < , 2≤ ≤ −1.
or there exist a sequence of elements = , = , , …, = , = in such that
( )− ( ) = | ( )− ( )| < , 1≤ ≤ .
Hence the result.
Theorem 3.4: Let ( , , ) be a bitopological space and = ∩ . Let be a dense subset of a topological space
( , ) and for every ε> 0 and let there exist a continuous function ∶( , ) →[0,∞) such that is function
– – – chainable. Then the bitopological space is function – − – – chainable.
Proof: Follows from Theorem 3.3
Theorem 3.5: Let ( , , ) and ( , , ) be two bitopological spaces and = ∩ and = ∩ . Let be an onto continuous function from topological space ( , ) to topological space ( , ). Then if the bitopological space is function – − – ε – chainable, the bitopological space is function – − – ε – chainable.
Proof: Let , ∈ .Then there is a sequence of points ( ) = , , … , = ( ) in a bitopological space such that | ( )− ( )| < ; 1≤ ≤ .
or there is a sequence of points = , , … , = ′ in the bitopological space such that ( ) = ; 1≤ ≤
and | ( )− ( )| < ; 1≤ ≤ .
Hence bitopological space ( , , ) is function – − – ε – chainable.
Theorem 3.6: Let ∶( , )→( , ) be bijective open map where ( , , ) and ( , , ) are two bitopological spaces and = ∩ , = ∩ . If the bitopological space ( , , ) is function – − –ε– chainable then the bitopological space ( , , )is function – − – ε – chainable.
Proof: Let ( ), ( )∈ where , ∈ . Now there exists a sequence of points = , , … , = ′ in such that | ( )− ( )| < ; 1≤ ≤ .
or there is a sequence of points ( ) = ( ), ( ), … , ( ) = ( ) in such that
( ( ))− ( ) = | ( )− ( )| < ; 1≤ ≤ .
or is function – − – ε – chainable.
Theorem 3.7: Let ( , , ) and ( , , ) be two bitopological spaces and = ∩ and = ∩ . Let
∶( , )→( , ) be a homeomorphism. Then the bitopological space ( , , ) is function – − – ε – chainable
iff the bitopological space ( , , )isfunction – − – ε – chainable.
Proof: Follows from Theorem 3.5 and 3.6.
Theorem 3.8: Let ( , , ) and ( , , ) be two bitopological spaces and = ∩ and = ∩ . Let be an injective open map from a topological space ( , ) to a topological space ( , ). Let be any subset of and ( ) =
. If the bitopological subspace is function – – ε – chainable then the bitopological subspace of ( , , ) is function – − – ε – chainable.
Proof: Follows from Problem 24, chapter 7 [6] and Theorem 3.6.
Theorem 3.9: Let ( , , ) and ( , , ) be two bitopological spaces and = ∩ and = ∩ . Let be an homeomorphism from a topological space ( , ) to a topological space ( , ). Let be any subset of and ( ) =
.Then bitopological subspace is function – – ε – chainable iff the bitopological subspace is function – −
Proof: Follows from Problem 25, chap. 7 [6] and Theorem 3.9.
Theorem 3.10: Let ( , , ) be a bitopological space and ( , ) is a topological space where = ∩ and let
= ∪ where and are closed sets in topological space . Let be function – − – ε – chainable and be function – − – ε – chainable such that ( ) = ( )for every ∈ ∩ . Then the bitopological space is function – − ℎ – ε – chainable where
ℎ( ) = ( ), ∈
( ), ∈
Proof: By pasting lemma the function ℎ ∶ →[0,∞) is continuous.
Then the bitopological space is function – − ℎ – ε – chainable follows directly from definition of ℎ and function – – ε – chainability of and function – − – ε – chainability of .
And hence the bitopological space X is function – − ℎ – ε – chainable.
Theorem 3.11: Let ( , , ) be a bitopological space, = ∩ and , ⊂ such that ~ and ~ are separated sets and = ∪ . Let for every ε> 0 thereexist a function : ( , )→[0,∞)such that ∶ →[0,∞) and ∶ →[0,∞) are continuous. If is function – − – – chainable and is function – − – –
chainable then bitopological space X is function – − – – chainable.
Proof: Now : →[0,∞) and : →[0,∞) are continuous functions. By Problem B, chap. 3[5] , is continuous on .
Again ( ) = ( ), ∈
( ), ∈ and ( ) = ( ) for every ∈ ∩ .
Then by Theorem 3.10, is function – − – ε – chainable.
Theorem 3.12: Let ( , , ) and ( , , ) be two bitopological spaces and = ∩ and = ∩ .Let ( , )
be a compact space and ( , ) be a Hausdorff space and ∶( , )→( , ) be a continuous bijection. Then bitopological space is function – − – ε – chainable iff bitopological space is function – − – ε – chainable.
Proof: Now is a homeomorphism by Corollary 2.4, chap. 7[4]. The result then follows from Theorem 3.7.
Theorem 3.13: Let ( , , ) be a bitopological space, = ∩ and { }be a sequence of continuous functions from ( , )to [0,∞) such that { } uniformly converges to a function ∶( , ) →[0,∞). If bitopological space is function – – chainable for each ∈ ℕ then bitopological space is function – − – chainable.
Proof: Now ∶ →[0,∞) is continuous, by Theorem 4.4 [4].
By uniform convergence; there is ∈ ℕ such that| ( )− ( )| < for all ∈ and for all ≥ . Let ≥ and let , ∈ .
Then there is a sequence of points = , , … , = such that | ( )− ( )| < ; 1≤ ≤ .
Also | ( )− ( )| < ; and | ( )− ( )| < .
Consequently, | ( )− ( )| < 3 ; 1≤ ≤ .
Hence the result.
Theorem 3.14: Let ( , , ) be a bitopological space, = ∩ . For every > 0, a normal space bitopological space is function – − − – chainable for some function on .
Proof: Choose two non -negative real numbers and such that − < .
Let and be disjoint closed subsets of .
By Urysohns Lemma, there exists a continuous function ∶( , ) →[ , ] where ( ) = for all ∈ and ( ) =
for all ∈ .
or | ( )− ( )|≤ − < for all , ∈ .
Theorem 3.15: Let ( , , ) be a bitopological space and = ∩ . Let ( , ) be a compact uniformly function –
− – chainable space for some positive real valued function on . Then there exists a positive real number such that
( ) + 1 > ( )
( )forsome , ∈ .
Proof: Let , ∈ such that
( ) = inf
∈ ( ) and ( ) = sup∈ ( )
By Problem A(b) and (c), chap. 5 [5], there is an > 0 such that ( ) > for all ∈ .
Now there is a sequence of elements = , , … , = in X
with| ( )− ( )| < ≤ ( )or ( )− ( ) < ( )
Putting ( ) = ( ) for some > 1, ( ) < ( )≰
Hence ( ) + 1 > ( ) ( ) .
EXAMPLES
1. Let ( , , ) be a bitoplogical space where X is the set of natural numbers, is a discrete topology on X and is an odd-even topology which is partition topology generated by = {1,2}, {3,4}, {5,6}, … on X and
= ∩
(i) Let ∶( , )→[0,∞) defined by (2 ) = , (2 −1) = is continuous.
Then the bitopological space is function – − – ε – chainable for > 1.
(ii) Let ∶ →[0,∞) defined by (2 ) = 1 , (2 −1) = 1 is continuous. Then the bitopological space is function – − – ε – chainable for > 0.5.
(iii) Let > 0 choose ∈ ℕ with > 1 .
Define ∶ →[0,∞) by (2 ) = 1 , (2 −1) = 1 is continuous. Then the bitopological space is function – − – ε – chainable for any > 0.
2. Let ( , , ) be a bitoplogical space with and be two partition topology on space ⊂[0,∞) ×ℝ
generated by the sets = {( −1, ): ∈ ℝ }; ≥1 and = {( + 1, ): ∈ ℝ }; ≥ −1 . Then the bitopological space is function – − – chainable where is the projection map given by
( , ) = ; ( , )∈[0,∞) ×ℝ.
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