J. Comp. & Math. Sci. Vol. 1(2), 197-199 (2010).
On Weakly Urysohn Spaces S. S. BENCHALLI* and G. P. SIDDAPUR** Department of Mathematics, Karnatak University, Dharwad-580 003 Karnataka ( India) Email: benchalliss@gmail.com*, siddapur_ math@yahoo.com** ABSTRACT A topological space X is said to be weakly Urysohn if for each x, y in X with distinct closures there are neighbourhoods U and V of x and y respectively such that the closures of U and V are disjoint. In this paper a characterization of R0-spaces is given. It is noticed that a regular R0-space is weakly Urysohn and a weakly Urysohn space is R1. Also a weakly Urysohn T0-space is a Urysohn space.
2000 Mathematics Subject Classification: 54A05. Key words: R0-spaces, R1-spaces, Urysohn Spaces.
1 INTRODUCTION A topological space X is said to be a Urysohn space1 if for each pair of distinct points x, y in X there are neighbourhoods U, V of x and y respectively such that U V = . It is known that every regular T1-space is Urysohn and that every Urysohn space is Hausdorff. In the present note a weaker form of Urysohn spaces is proposed. It is observed that a regular R0-space is weakly Urysohn and a weakly Urysohn space is R 1. Also a weakly Urysohn T0-space is a Urysohn space. If X is any topological space and x X then
x stands for the closure of {x}. 2 Main Results 2.1 Definition A topological space X is said to be weakly Urysohn if for each pair of points x, y in X with
x y there are neighbourhoods U and V of x and y respectively such that U V = . Every Urysohn space is weakly Urysohn, but not conversely. 2.2 Example Let X={a, b, c} and
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
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S. S. Benchalli et al., J.Comp.&Math.Sci. Vol.1(2), 197-199 (2010).
={X, , {a}, {b, c}} be a topology on
x X y . From (a) it follows that x
X. Then {X, , {b, c}, {a}} is the family of all closed sets. Note that X is a weakly
X y . Therefore x y = and hence
Urysohn space. We have a ={a}, b =
y x . Thus (c) holds..
{b, c} and c = {b, c}. Now a b . Let et U = {a} and V = {b, c}. Clearly U and V are neighbourhoods of a and b
(c) (a): Suppose (c) holds. Let G be an open set in X and x G. Then x
respectively such that U V = { a } { b, c }= {a} {b, c} = . Also a c . Let U = {a} and V = {b, c}. Then U and V are neighbourhoods of a and c respectively such that U V = . Therefore X is weakly Urysohn. X is not a Urysohn Space; For b, cX with bc there do not exist neighbourhoods U and V of b and c respectively with
U V =. Therefore X is not Urysohn. 2.3 Theorem The following statements about a topological space X are equivalent. (a) For each open set G in X, if x G then x G. (b) For each x, y X either x = y or
x y =. (c) For each x, y X if x y then y
x.
X G. Let y X G. Then y X G. Therefore x y . From (c), y x . Therefore y X x . Thus X G X x . Therefore x G. Thus (a) holds. Thus (a) and (c) are equivalent. The equivalence of (b) and (a) is proved in2. 2.4 Definition 1 A topological space X is said to be R0 if for each x, y X, x y implies y x . 2.5 Definition 3 A topological space X is said to be R1 if for every pair of points x, y of X with x y implies x and y have disjoint neighbourhoods. 2.6 Theorem The following statements about a topological space X are true.
holds. Let x, y X and x y . Then x
(a) A regular R0-space is weakly Urysohn. (b) A weakly Urysohn space is R1. (c) A weakly Urysohn T 0 -space is Urysohn.
X y . Now X y is an open set and
Proof. (a) Let X be regular R0-
Proof. (a) (c): Suppose (a)
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
S. S. Benchalli et al., J.Comp.&Math.Sci. Vol.1(2), 197-199 (2010). space. Let x, y X with x y . From om
199
hoods U and V of x and y respectively
Theorem 2.3 (b), x y = which implies
such that U V =. Hence X is a Urysohn space.
that x y . Therefore there are open sets U(x) and V (y) containing x and y respectively such that U(x) V (y)=.
ACKNOWLEDGEMENTS
Therefore U(x) V (y)=. Now w x U(x) and X is regular. Therefore there exists an open set W containing x such that x W W U(x). Therefore
W V (y)=. Hence X is weakly
The authors are grateful to the University Grants Commission, New Delhi, India for its financial support under UGC SAP I to the Department of Mathematics, Karnatak University, Dharwad, India and also for UGC-Research Fellowship under RFSMS Scheme, to the second author.
Urysohn. REFERENCES (b) Let X be weakly Urysohn. Let x, y X with x y. Then by definition there are neighbourhoods U and V of x and y respectively such that U V =. Therefore U V =. Hence X is a R1-space. (c) Let X be a weakly Urysohn T0-space. Let x, y X with x y. Since X is T0,
x y . Therefore there are neighbour-
1. K. Csaszar, New Results on Separation Axioms, Proc. Intern. Symp. On Topology and Its Appl. Herceg-Novi (Yugoslavia), 25-31.8.1968 p. 118120. 2. A. S. Davis, Indexed Systems of neighbourhoods for topological spaces, Amer Math Monthly, 68, 886-893 (1961). 3. M.G. Murdeshwar and S.A. Naimpally, R1-Topological Spaces, Canadian Math Bull Vol. 9, 521-523 (1966).
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