Characterization of Function-Ε-Chainble Sets in Topological Space

7/28/2019 Characterization of Function--Chainble Sets in Topological Space

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Mathematical Theory and Modeling www.iiste.org

ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)

Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications

189

Characterization of Function--Chainble Sets in Topological

Space

PRIYA CHOUDHARY, VIJETA IYER and KIRAN SHRIVASTAVADepartment of Mathematics, S.N.G.G.P.G. College, Bhopal (INDIA)

Mailing address :C- 473, Shahpura Bhopal-462016 (INDIA)Mailing address :C-478,Shahpura Bhopal-462016(INDIA)

D/79,Nehru Nagar Bhopal-462003 (INDIA)email id:[email protected] id: [email protected]

email id:[email protected]

Abstract

In this paper we introduce the concept of function--chain between two sets in topological spaces throughcontinuous function which is the extension of function--chain between two points of the space. Simplecharacterization of function--chainable sets in terms of function -chains between their points has been

established. In case of metric space, the equivalence of -chainability and function--chainability of sets is alsoestablished in this paper. Further some results of [1] have been generalized.Subject Classification: AMS (2000):54A99

Keywords: Simple chain, chain between two points, chain between two sets, chainable metric space.Throughout this paper will stand for topological space with topology and [0, ) will be a realvalued non-constant continuous function unless stated otherwise.1. Definitions

Let be a subset of the topological space. For > 0,let () = { : () () < }, where () () = { () () : }

1.1. Definition

Let .Then diameter of is defined to be !"{ () () , } and is denoted by #().

1.2. Definition

Let, $ . Then-distance between and $ is defined to be { () (%) , % $} and isdenoted by &(, $).

1.3. Remark

() = {: &(, ) < }

1.4. Definition

A topological space (, ) is said to be function- -chainable if for > 0 there exists a non-constantcontinuous function: [0, ) such that for every pair of elements , there is a sequence

= , *, +, , - = of elements in with ( ) ( /*) < ; 1 1.5. Definition

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 function chainable.1.6. Definition

Let , $ . A function chain of length from to $ is a finite sequence ,, *, +, , -ofsubsets of with = , - = $, /* () and (/*) . If function chain exist

between and $ we say that < , $ > is function chainable and < , $ > is function chainable ifit is function chainable for each positive .

Using the notation inductively construct the set - () for each 45 as follows:

* () = () for each 2 set

- () = ( -/*()). The following should be observed:

(1) - ()

-5*()

(2) - () -()

We set 8 (< , $ >) to be the length of the shortest function chain between and $.

1.7. Example of function 9 chainable setsLet be a topological space with odd even topology which is a partition topology generated by ={ {1,2}, {3,} , {,6}, . } and : [0, ) define by (2@) = @ , (2@ 1) = @ is continuous function. Let

= {1,2} , $ = {3,} and = 1.2 then () = {1,2,3,} and ($) = {1,2,3,,,6} or ($) & $ () AB = , * = $ then < , $ > is chainable for = 1.2


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2. Theorems

2.1. Some results whose proofs are obvious hence omitted.2.1.1.ResultLet, $ , then

i. &(, $) < if$ () ii. &(, $) < if ($) iii. ()iv.iv.iv.iv. () ($) if $v. () ($) = ( $)

vi. ( $) () ($)2.1.2. ResultIf < , $ > and < J , K > are chainable then < J , $ K > is also chainable where

, $, J, K 2.2. Theorem

Let (,)be a topological space and then

L

() =

Proof : s () , > 0 then L ()

Let then () () () or there exists such that |() ()| < or (), > 0

or L

()

Suppose that L

() or there exist L

() such that

or there exist (), > 0 such that and hence

Or|() ()| 0 or |() ()| = for some real number > 0 or () for < .

This contradicts that () > 0

Hence = L

()

2.2.1. Collolary

is closed if and only if =

L ().

Characterization of function chainable sets in terms of function chains between points andsequence is given below.2.3. Theorem

Let, $ and < , $ > be function chain from every point of to some point of$ and vice-versa .Also converse holds.Proof: We prove the necessary part first. As < , $ > is function chainable there exists a sequence, * , , - of subsets of with = , - = $, (/*) and/* () ; 1 . Let

be arbitrary. Then or (*) AB |() (*)| < for some * * .

Again * then |(*) ( +)| < for some + +. Repeating the above process times we obtaina sequence of points = , *, +, , - = $ such that |() ( /*)| < ; 1 and

, showing that there exist a function chain from to . Likewise we can obtain a function chain from every point of$ to a point of.We next prove the sufficient part. Let there exist a function chain from every point of to

some point of$ and vice-versa. Let * = { : |() ()| < for some and }. Clearly* and * (). Next we show that (*) . If then there exist a sequence

= , *, + , - = $ such that |() (*)| < or * * then |() (*) | < or (*) or (*).

Again let + = { : |() ()| < AB A * & }.Clearly+ ,+ (*) and it can be shown as above that * (+). Repeating the above process

times we obtain a sequence = , * , , - = $ of subsets of , such that < , $ > is function chainable.2.4. Theorem

Let, $ , if #( $ ) , then < , $ > is function chainable.


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Proof : ObviousIn next theorem the equivalence of chainability and function- chainability of two sets is obtained inmetric spaces.2.5. Theorem

Let (, &) be a metric space and (, )be topological space. If< , $ > is -chainable then < , $ > is chainable for some continuous function [0, ).Proof: Let and [0, ) be defined as () = &(, ) . Let < , $ > be

chainable where, $ . Then there exist chain of finite sequence = , *, +, , - = $ such that

/* () and (/*). Let x () or &(, ) < AB inf

g&(, ) < or d(x, a) < for some .

Now &(,) &(, ) &(,)or () () &(, ) < AB |() ()| < or|() () | < or () or () ().

Hence /* () and (/*) or < , $ > is chainable.

2.6. Theorem

Let < - > be monotonically increasing sequence of positive real number converging to (arbitrary). Then

< , $ > is function -chainable if and only if there exists a subsequence < -k > of< - > such that< , $ > is function -k-chainable for each @ l.

Proof: Similar to proof of theorem 2[1]2.7. Theorem

Let , $ . If ( $ ) is connected and > {#(), #($)} then < , $ > is function

chainable.Proof: Similar to proof of theorem 4[1]2.8. Theorem

Let , $ and > {#(), #($), &(,$)} then < , $ > is function chainable and

8(< , $ >) = 2.

Proof :Similar to proof of Preposition [1]2.9. Theorem

is function chainable if and only if< , $ > is function chainable for every pair of subsets, $ of.Proof : Similar to proof of theorem 5[1]2.10. Theorem

Let, $ . Then = $ if and only if< , $ > is function chainable and 8 (< , $ >) = 1.

Proof : Similar to theorem 7[1]2.11. Theorem

If - () $

-5*() , then < , $ > is function chainable and 8 (< , $ >) = 1 .

Proof : Similar to proof of theorem 8[1]2.12. Theorem

Let be function chainable. Define a relation ~ on as follows:< , $ > ~ < J, K > if and only if8 (< , $ >) = 8 (< J , K >).

Then ~ is an equivalence relation on , which partitions into disjoint equivalence classes denoted

by < , $ > , < J, K > .Proof : Obvious.2.13. Theorem

Let { = , *, +, , - = $} be a simple chain [3] then < , $ > is function 2 chainable where > {#(), #(*), #(+), , #($)}.

Proof : Let , ( *), *then |() ()| < and |() ()| < or|() ()| < 2 or |() (*)| < 2or +(*) or +(*).

then inf

|() ()| < 2 AB |() ()| < 2

Or +() or +(*) and * +().

Similarly* (+), + (*), , -/* (-) and- (-/*).

Thus the sets *, +, , -/* forms a function chain from to $ that is < , $ > is function 2 chainable.


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REFERENCES1. SHRIVASTAVA KIRAN, AGRAWAL GEETA, Characterization of -chainable sets in metric spaces,

Indian J. Pure Appl. Math. 33(6):933-940, June 2002.2. MUNKERS, James R, Topology, A first Course PHI, 1987.3.

LIPSCHUTZ, S., Schaums outline of theory and problems of General Topology, 1965.4. KELLY, J.L., General Topology, Van Nostrand Reinhold Company, New York, 1969.

5. JOSHI, K.D., Introduction to General Topology, Wiley Eastern Limited, 1992.6. STEEN, LYNN ARTHUR; SEEBACH, J. ARTHUR JR.(1995)[1978], Counterexamples in

topology (Dover reprint of 1978 ed.), Berlin, New York: Springer- Verlag.7. IYER VIJETA, SHRIVASTAVA KIRAN, CHOUDHARY PRIYA, Chainability in topological spaces

through continuous functions, (accepted).


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Characterization of Function-Ε-Chainble Sets in Topological Space

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