CERTAINFUNDAMENTALPROPERTIESOF a -OPENSETS IN TOPOLOGICAL SPACES
Dr.S.Bamini', Assistant Professor, Department of Mathematics, G.Katheeja Mary2’ Assistant Professor, Department of Mathematics, K.Su1ochana3 Assistant Professor, Department of Mathematics, Marudhar Kesari Jain College for women, Vaniyambadi.
ABSTRACT:
In this paper, we are going to see a new type of an n-sets called e”-opensets.And we also discuss here about some properties of o * - open sets and we compare this new notation with other some classes of sets.Then some of these characterizations,remarks,andcounterexamplesaredetailly given in this work.
Keywords:
o-open sets,a-regular spaces, o-continuity mappings,o'-open sets, o”-opensets
Introduction
This notionof set is one of the most important concepts in the topological space,and itplays an vital role in the topic of general topology.There are various topics studied on setstheory and their applications in soft set, fuzzy sets and orher see([1]-[3]).An a-open set isone of these sets see([4],[5]).This topic was studied first by O.Njasted in 1965.These selsformthetopologyonX whichis finertharrr,see([6],[7]).GovindappaNavalagi,hedefinedtheconcept of semi o-open sets by considering o-open sets instead of open see 8], for moredetailed concept of this topic see([7],[9]). Generally the aim of this paper is to introduce andinvestigate thenew class of sets as an extension ofan o-sets called any -open sets and it isstrlctly weaker than n-open sets and stronger than semi tt-open sets. Many exciting andsomeimportant results are given.In thls Study (X,i) or a space X denote a topological space onwhich no separatlon axioms are assumed unless explicity stated.The closure (resp.,interior)ofasetUof spaceXwillbedenotedas cl(A) (resp.,int(A)).
2. Preliminaries
A set A is an n-open, if A G int cl(int{A)) and its complement is an ‹f-closed [6]. Also, it is semi n-open if A ñ cl(int(cl pint A))) or equivalently Adcl(oint (A))and its complement is semi a-closed st [7]. The family of all n- open set (tf-closed, semi ff-open, semi a-closed) is denoted by BO(X) (resp.,) 8C(X),s8O(X),s8C(X).
A set A is an n-open, if A G int(cl int(A)) and its complement is am-closed, it is semi n-open if A G
ci(int(ci(inf(A))) or equivalently A ñ cl(8int(A)) and its complement is semi n-closed set. The family of all a-open set (a-closed, semi tr-open, semi a-closed) is denoted by,O(X) (resp., qC(X), ,qO(X), „C(X).
The union (resp., intersection) for each n -open set (resp., n-closed) sets contained in (resp., contained) A is
named an
n-interior (resp., and o-closure ) of A denoted by mlnf(fi)(resp., acl(A)).
We have:int(A)G int(A) incl(A)Gc1(A).
A space A is called hyperconnected if every non empty open set of X is dense.
Certain Fundamental Properties af a”-Open sets.
Now we will discussed about class of sets called an n"-open as an extension of an n‘-open sets in topological
space.
Definition
A set A of a space A is called an o**-open if
A j oint(cl(8int(A))), where its complement is called an ff**-closed. The family
of all a*-open (resp., a*-closed) is denoted by a *
O(A)(resp., a*C X)).
Example:
Assume that X = (a,b,c,d,e,f}withz—{X,(b},(d},(e),(f},(b,d},{e,f),(b,d,f},p},O(X)—( (a,b,d},(b,c,d},(c,d,f},(d,e,f}} U
‹. Then we can assume U=(c ) then UUaint(cl(gint(c1(oint(U)),thusUisn”-open set.
Remarks
1.
ii.
111.
lV.
Candy are n*-open set in every space N.
In the Discrete space A, then n*O (A) - P(A), where P(X) is power set of a space N. If a space A is Indiscrete space, the A anda are the only n*-open sets.
In the usual topological space, every open interval is an n* -open set.
Proof.’ Obvious.
Thearem{1)
Let A be a space. Ten:
i.
Every open set A in a space N
(resp., n-open set) is a n*-open set.
ii.
Every n*-open is semi n-open set.
Proof.’ (i)
Assume that A is
open, thus A is n-open
Hence,A-gint(A) so
int(A)Gcl(«int(A))
int(pint(A))=oint(A) joint(cl(eint(A)))
dcl(int(cl(int(A))))
hint(cl(int(cl(int(A)))))
ie)
Thus A Cq • 0(X).
Now
assume that A is a*-open set, then
Añc1(oint(A)) But
Int(cl(int(A)))
Paint(cl(int(A)))
G int(ci(Int(A))
joint(cl(aint(cl(eint(A))))) Thus A Cq 0(X)
Proof.’ (ii)
Assume that A
C p • O (X)
WAGoint(cl(oint(c1(oint(a)))))
But
aint(cl(pint(cl(aint(A)))))Gcl(eint(cl(«int(A))))
joint(cl(pint(A))) dcl(eint(A))
ie) Adcl(oint(A)).
A e sa”0(X).
Thearem{2)
A o ••
0(X) Wthere
existsB Cq • 0(A) such that B
GAG«int(c1(eint(cl(B)))).
Proof.’
(Neces.sit y )
Assume that A
C q-• 0 (X)
A Koint(cl(oint(cl(A))))
'cl(eint (cl(eint (A)GA, aint (cl(A))is an a“-open set Put
B=oint(cl( int(A)))
GB GAGaint(cl(eint(cl(B))))
Sufficienc y:
Suppose there exists n**-open set B such that BGA G8int(c1(oint(cl(B)))) veint(cl(oint(A))))is
largest a**-open sett A
B Paint(cl(pint(cl(A))))
B jeint(cl(pint(B)))GA
joint(cl(pint(cl(A))))
G0int(cl( Oint(cl( «int(A))))
A
joint(cl(pint(cl(eint(A))))
A C q••
0 X ) Theorem(3)
A Cq • C(X) Wthere exists C
8 C (X ) such that
cl(oint(cl(oint(F))))GAGF
Proof:
Proof.’ (Nece.s.sit y)
Assume that A Cq •• 0 (X)
Añocl(int(dcl(int(A))))
'int(ecl (int(ccl (A)GA,ecl (int(A)) is an cr**-open
set
ie)
Put B= cl(int(ccl(A)))
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