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عنوان البحث(Papers / Research Title)

## Separation Axioms Via Kernel Set in Topological Spaces

الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)

لؤي عبد الهاني جبار السويدي

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لؤي,عبد,الهاني,جبار,السويدي ,Separation Axioms Via Kernel Set in Topological Spaces , Time 14/12/2016 09:00:39 : كلية التربية للعلوم الصرفة

وصف الابستركت (Abstract)

In this paper deals with the relation between the separation axioms Ti-space , i= 0,1,…,4 and Rispace i= 0,1,2,3 throughout kernel set associated with the closed set .

الوصف الكامل (Full Abstract)

INTRODUCTION AND PRELIMINARIES
In1943, N.A.Shainin [4] offered a new weak separation axiom called R0 to the world of the
general topology. In 1961, A.S.Davis [1] rediscovered this axiom and he gave several interesting
characterizations of it. He defined R0, R1 and R2 entirely. He did not submit clear definition of R3-
space but stated it throughout this note: ( But the usual definition of “normality” must be modified
slightly if R3 is to be the axiom for normal spaces.)
The present study presents the definition of R3-spaces as follows:(A topological space is
called an R3-space iff it is normal space and R1-space). This definition of R3-space satisfied with: Every
R3 is an R2-spaces. On the other hand (X, T) is aT4-space if and only if it is an R3 -space and ???? ?1-
space, ?? = 0, 1, 2, 3,4.
We proved Ri-spaces, ?? = 0,1,2,3, by using kernel set[2,5] associated with the closed set.
We prove the topological space is aT0-space if and only if either ?? ? ??????{??} or ?? ? ker?{??} for
each ?? ? ?? ? ?? .and a topological space (X,T) is a T1-space if and only if for each ?? ? ?? ? ?? ,then
?? ? ker?{??} and ?? ? ker?{??}, also (X,T) is a T1-spacce iff ??????{??} = {??}, and by using kernel set, we
states the relation between Ti-spaces ?? = 0,1,2,3,4 and Ri-spaces ?? = 0,1,2,3.

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