ON θ Generalized Pre- Open sets in a Topological Space

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ON θ Generalized Pre- Open sets in a Topological Space

Ms. Sathya, Ms. Aruna

Assistant Professor in Mathematics,

Thassim Beevi Abdul Kader College for women and Mohamed Sathak Hamid College of Arts and Science College Affiliated to Alagappa University, Karaikudi, Tamilnadu, India.

Abstract:- In this paper, a new class of sets called theta generalized pre-open set in a topological space introduced and some of their basic properties are investigated. Several examples are provided to illustrate the behaviour of new sets.

Keywords: Pre- closed set, Pre open set, gp -Closed, gp open set.

  1. INTRODUCTIONN.V.Velico and J.Dontchev etal are introduced the concepts of -genaralized closed sets. The concept of generalized closed sets introduced by Levin plays a significant role in topology. After the introduction of generalized closed sets many research papers were published which deal with different types of generalized closed sets. H.Maki et al. defined the concept of gp-closed set in topological spaces and established results related to it. These concepts motivated us to define a new class of sets called the theta generalized pre-closed sets and gp-open sets.
  2. GENERALIZED PRE -OPEN SETS.

Definitions 2.1: A subset A of a topological space X is called a Theta generalized pre open ( briefly, gp open ) set if is

gp-closed.

Example: Let x={ a, b, c, } and topology = { x, , { a} } then gp- open set = { x, , {a}, {c}, {b}, {a,b} {a,c} }

Theorem 2.2 A set A X is gp open iff G pint(A) whenever G is closed and G A.

Proof: Let A be gp open set and suppose G A where G is -closed. Then X-A is a gp closed set contained in the open set X-G, pcl (X-A) X-G .

Since pcl ( X-A ) = X Pint ( A) [ 9 ], then X pint ( A ) X-G. That is Gpint(A).

Conversely, let Gpint( A ) be true whenever G A and G is closed, then X-pint (A) X-G. that is pcl (X-A) X-G. this implies X-A is gp closed and A is gp open in X.

Example 2.3: Let X={ a, b, c,} and topology = { x, , {c}, {a, c} } then gp- open set = { x, , {a,c}, { a.b}, {c}, {a} } . the converse of the above theorem need not be true.

Theorem 2.4 If A is gp open and B is any set in X such that pint (A) BA, then B is gp open in X.

Proof: Follows from the definitions and theorem 3.8 [ 7 ]

Theorem 2.5 If A is gp open and B is any set in X such that pint (A)B, then AB is gp open in X.

Proof: Let A be a gp open set of X and pint(A)B, then A pint(A) ABA.

Since pint(A) A, then pint(A) ABA and from theorem 2.4, AB is gp open in X.

Theorem 2.6 If a set AX is gp-closed, then pcl (A) -A is gp open in X .

Proof: suppose that A is gp-closed and M is closed such that M pcl (A) A, then by theorem 3.5 [7], M = and hence M pint( pcl (A) -A). Therefore by Theorem 2.2pcl(A) – A is gp open.

Example 2.7 Let X= { a, b, c} and topology ={ x, , {a}, {a,b}, }, then gp open x = { x , , {a,c}, {a,b}, {c}, {a}, {b},}.

Definitions 2.8 [3] Let A and B be two non void subsets, of a topological space X. Then A and B are said to be separated if A cl(B) = cl(A) B = .

Theorem 2.9 If A and B are separated gp open sets, then AB is gp open.

Proof: Let F be a closed subset of AB. Then F cl (A) (AB) cl(A)= (A cl(A))(Bcl(A))=A =A. That is, F cl (A) A. Therefore Then F cl (A) is a closed set contained in A and A is a gp open, then by Theorem 2.2, F

cl (A) pint(A). similarly F cl (B)pint(B). Thus we have F = F(AB)=(F A) (FB) (F cl (A)) (F

cl (B)) pint(A)pint(B)pint(AB). That is Fpint (AB). Hence by Theorem 2.2, AB is gp open.

Related Nbhds, closure and Interior.

Definition 3.0 a subset M of a topological space X is called gp -neighbouhood (briefly , gp -nbhd) of a point xX, if there exixts a gp-open set U such that xUM.

The collection of all gp -nbhds of a point xX is called gb -nbhd system of x and is denoted by gpN(x).

Theorem 3.1 If A is gp -open set ,then it is gp -nbhd of each of its points .

Proof: let A be any gp -open set of of X, then for each x A, xAA. Therefore A is gp -nbhd of each of its points.

Theorem 3.2 if AX is a gp -closed set and x , then there exists a gp -nbhd F of x such that FA=.

Proof : Let AX is a gp -closed set, then is gp -open. Therefore, By theorem 3.0, is gp -nbhd of each of its points. Let x then there exists a gp -open set F such that x . That is, FA=.

Theorem 3.3 Let x be a point in a space X, then

(i) gpN(x) .

  1. If A gpN(x),then x A.
  2. If A gpN(x) and B A ,then B gpN(x).
  3. if gpN(x) for each then gpN(x).

Proof (i) since X gpN(x), gpN(x) .

  1. let A gpN(x ), then there exists a gp-open set G such that xGA. This implies xA.
  2. Let A gpN(x ), then there exists a gp-open set G such that xGA. Since AB, then xGB. This shows BgpN(x ).
  3. since for each , is gp -nbhd of x, then there exists a gp-open set such that x . Which implies that x and hence gpN(x).Theorem 3.4 Let A be a subset of a X. Then () if and only if , for every set U containing .Proof:Let (). Suppose that there exisits a set U containing x such that , then and

    is . Therefore () , which implies (), a contradiction.

    Conversely, suppose that (). Then there exists a set containing A such that . Hence is a

    set containing . Therefore = , which contradicts the hypothesis.

    REFERENCE:

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    4. H. Maki, R. Devi and K. Balachandran, generalised -closed sets in topology, Bull. Fukuoka Univ., Ed., part III, 42 (1993),13-21.
    5. H. Maki ,J. Umehara and T. Noiri ,Every topological space is pre – 1/2 Mem.Fac.Sci.Kochi.Univ.Math.,17(1996),33-42
    6. J . Dontchev and H. Maki ,-generalized closed sets ,Internat.J.Math.Sci.Vol.22,No.2(1999)239-249
    7. J. Cao ,M. Ganster and I. Reilly ,on preclosed sets their generalizations ,Houston J.Math.,28(2002),771-780
    8. C.Aruna and R.Sathya, on generalized pre closed sets in topological spaces, www.ijar.org (E -ISSN 2348-1269, P -ISSN 2349-5138) march 2019

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