H03702061062

The International Journal Of Engineering And Science (IJES) || Volume || 3 || Issue || 7 || Pages || 61-62|| 2014 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805

On Uniformly Continuous Uniformity On A Topological Space 1

Dr. S.M.Padhye , 2Ku. S.B.Tadam 1,2

Shri R.L.T. College of Science, Akola

------------------------------------------------------------ABSTRACT:--------------------------------------------Given a topological space (X, ) we define uniformity on X such that (X, is uniformly continuous uniform space and it becomes the smallest uniformly continuous uniformity with respect to which set of continuous functions is larger than set of –continuous functions.

Keywords: Pseudo-metric space, Uniform space, Uniformly continuous function.

Subject code classification in accordance with AMS procedures: 54E15.

--------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 12 April 2014 Date of Publication: 20 July 2014 ---------------------------------------------------------------------------------------------------------------------------------------

INTRODUCTION Let X be any non empty set. Let be the family of all pseudo metrics on X. Then since discrete metric on X is a pseudo metric. For any subfamily of define the uniformity on X whose subbase is the family of all sets Vp,r = {(x,y) XxX /p(x,y)<r}, p

and r > 0

/

is a base for uniformity on X.

Now let (X, ) be a topological space. Let

(X) be the set of all continuous complex valued functions on X. For

every f (X), define df :XxX as df (x,y) = I f (x ) â&#x20AC;&#x201C; f( y) l. Then df is a pseudo -metric on X. With the help of pseudo-metric df on X define an open sphere Sr(x), x X, r > 0 as Sr (x) = { y: df (x,y) < r } As the intersection of any two open spheres contains an open sphere about each of its pts , the family { Sr (x) / x X } is a base for topology for X. This topology is the pseudo-metric topology for X generated by pseudometric df on X say .

Let P be the family of pseudo metrics df on X as f (X). Then P defines a unique uniformity on X such that S = { Vf,r / f (X) and r > 0 } forms a subbase where Vf,r = {(x,y) XxX / df (x,y) < r} We denote this uniformity on X by Definition: Uniformly continuous uniform space: A uniform space is said to be Uniformly continuous uniform space if every real valued continuous function is uniformly continuous.

Theorem1: Let (X, ) be a topological space and be the uniformity defined on X as above then (X, ) is a Uniformly continuous space.

Proof: For proving this theorem we require following two lemmas.

Lemma1. Every

-continuous mapping is

-continuous.

Proof: Let f: X be any -continuous mapping, where is the above defined uniformity on X. Then we show that f is -continuous. Let G be any open set in . Then f-1(G) is â&#x20AC;&#x201C; open in X. i.e. where = {Tâ&#x160;&#x201A;X/ T U đ?&#x2019;° s.t. U[x]â&#x160;&#x201A;T} . Now we show that â&#x160;&#x2020; .

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On Uniformly Continuous Uniformityâ&#x20AC;Ś Suppose T

and

T. Then we have U

đ?&#x2019;° s.t. U[x]â&#x160;&#x201A; T. Since U

and r1,r2,â&#x20AC;Śâ&#x20AC;Ś.rn all positive such that But V[x] = { y: (x,y)

đ?&#x2019;° by defn of

. Then

f1,f2,â&#x20AC;Ś..fn

(X)

.

}

={ y: (x,y)

}

={ y: (x,y)

}

={ y:

}}

={ y: y

(

}}

={ y: y Now for each i, hence

is an open subset of is

and each fi is -continuous

â&#x20AC;&#x201C; open.

is â&#x20AC;&#x201C; open.

â&#x20AC;&#x201C; open set V[x] such that

is â&#x20AC;&#x201C; open ie.

This proves that T

. Hence

â&#x160;&#x2020; .

Lemma2: If f is â&#x20AC;&#x201C;continuous then f is đ?&#x2019;°-uniformly continuous function.

Proof: Since f is -continuous function, df(x,y)= l f(x)-f(y) l ,x,y X defines a pseudo metric on X and hence df P. Then for any r >0 Uf,r ={(x,y)/df(x,y)<r} is a subbase member of the uniformity đ?&#x2019;°. i.e. Uf,r đ?&#x2019;° such that (x,y) Uf,r l f(x)-f(y) l < r.Thus f is đ?&#x2019;°-uniformly continuous function.

- continuous complex valued function on X. By lemma1 f is â&#x20AC;&#x201C; â&#x20AC;&#x201C; uniformly continuous function. Hence (X, ) is uniformly continuous

Proof of Theorem 1: Suppose f is continuous. By Lemma 2 f is uniform space.

But from lemma2 every â&#x20AC;&#x201C;continuous function

Note: From lemma1 continuous and hence it is ie. â&#x160;&#x2020; = Theorem2:

â&#x20AC;&#x201C;continuous function.

Suppose (X, ) is a uniformly continuous uniform space such that

is the uniformity on X determined by â&#x20AC;&#x201C;continuous functions on X. i. uniform space such that â&#x160;&#x2020; Proof: To show that

- uniformly continuous

â&#x160;&#x201A;

where

U â&#x20AC;Śâ&#x20AC;Ś(1) .

â&#x20AC;&#x201C;continuous and

is a uniformly continuous uniform space, each fi is

. Hence for each . â&#x20AC;Ś

then

is the smallest uniformly continuous

such that W =

Since each â&#x20AC;&#x201C;continuous function fi is

Corollary:

â&#x160;&#x2020;

. Then there are â&#x20AC;&#x201C;continuous functions

â&#x160;&#x201A; , let U

f1,â&#x20AC;Ś.fn on X and

of uniform space) i.e.

is đ?&#x2019;°-uniformly

such that But

(by definition

is the smallest uniformly continuous uniform space such that

â&#x160;&#x2020;

Suppose (X, ) is a uniformly continuous uniform space such that then â&#x160;&#x201A; .â&#x20AC;Ś

References: [1].

[2].

[3].

J.L.Kelley, General Topology, Van Nostrand, Princeton, Toronto, Melbourne, London 1955.

W.J. Pervin, Foundations of General Topology,Academic Press Inc. New York, 1964.

Russell C. Walker, The Stone-Cech Compactification, Springer-Verlag Berlin Heidelberg New York 1974

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