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JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES
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ISSN 0976-5727 (Print)
ISSN 2319-8133 (Online)
Abbr:J.Comp.&Math.Sci.
2014, Vol.5(2): Pg.219-222
On Some New Open Functions in Topological Spaces
R. V. Pattanashetti
Department of Mathematics,
PC Jabin Science College, Hubli, Karnataka, INDIA.
(Received on: April 9, 2014)
ABSTRACT
A set A in a topological space (X , T) is said to be a g*p-closed
(briefly, g*p-closed) if pcl(A) ⊂ U whenever A ⊂ U and U is gopen in X. In this paper, we introduce quasi g*p-open map from a
space X into a space Y as the image of every g*p-open set is open.
Also, we obtain its characterizations and its basic properties.
Mathematics Subject classification: 54C10, 54C08, 54C05.
Keywords: Topological spaces, g*p-open set, g*p-closed set,
g*p-interior, g*p closure, quasi g*p-open function.
1. INTRODUCTION
The notion of pre-open set5 plays a
significant role in general topology. The
most important generalizations of regularity
(resp. normality) are the notions of preregularity1 and strong regularity (resp. prenormality6, strong normality6. Levine3 in
1963, started the study of generalized open
sets with the introduction of semi-open sets.
Then Njatsad7 studied a-open sets; In 1982,
Mashhour et al.5 introduced preopen sets
and pre-continuity in topology.
In 2002, Veerakumar12 have defined
the notion of g*p-closed sets, g*p-continuity
and g*p-irresolute maps. In this paper, we
will continue the study of related functions
by involving g*p-open sets. We introduce
and characterize the concept of quasi g*popen functions.
2. PRELIMINARIES
Throughout this paper, by spaces we
mean topological spaces on which no
separation axioms are assumed unless
otherwise mentioned and f: (X, τ)→(Y, a)
(or simply f: X→Y ) denotes a function f of a
space (X , τ) into a space (Y, a). For a subset
A of X, cl(A) and int(A) denote the closure
and interior of A, respectively.
2.1 Definition A subset A of X is said to be
pre-open5 if A ⊂int(cl(A)). The complement
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R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014)
of pre-open set is said to be pre-closed5. The
family of all pre-open sets (respectively preclosed sets) of (X , τ) is denoted by pO(X, τ)
[respectively pCL (X, τ)].
2.7 Definition A function f: (X, T) →(Y, σ)
is called a g*p-open function12 if the image f
(A) is g*p-open in (Y, σ) for each open set
A in (X, τ).
2.2 Definition Let A be a subset of X . Then
(i) pre-interior2,5 of A is the union of all
pre-open sets contained in A. (ii) preclosure2,5 of A is the intersection of all
pre-closed sets containing A.
The pre-interior [respectively preclosure] of A is denoted
by pint(A)
[respectively pcl(A)].
2.8 Definition A function f: (X, τ) →(Y, σ)
is called a g*p-continuous12 if the inverse
image of every closed set in (Y, σ) is g*pclosed in (X, T).
2.3 Definition A subset A of a space X is
called a g-closed set4 (briefly, g-closed) if cl
(A) ⊂U whenever A ⊂ U and U is open in
X. The complement of g-open set is said to
be g-open set4.
2.4 Definition A subset A of a space X is
called a g*p-closed set12 (briefly, g*pclosed) if pcl (A)⊂U whenever A ⊂ U and
U is g-open in X. The complement of g*popen set is said to be g*p-closed set8. The
family of all g*p-open sets (respectively
g*p-closed sets) of (X, T) is denoted by
G*pO (X, T) [respectively G*pCL(X, T)].
Definition 2.9. A function f: (X, T) →(Y, a)
is called a g*p-irresolute [12] if the inverse
image of every g*p-closed in (Y, a) is g*pclosed in (X, T).
3. QUASI g*p-open FUNCTIONS
3.1 Definition A function f: X→Y is said to
be quasi g*p-open if the image of every
g*p-open set in X is open in Y.
3.2 Theorem A function f: X →Y is quasi
g*p-open iff for every subset U of X, f(g*pint(U)) ⊂int(f(U)).
2.5 Definition Let A be a subset of X. Then
(i) g*p-fnterior12 of A is the union of all
g*p-open sets contained in A.
(ii) g*p-closure12 of A is the intersection of
all g*p-closed sets containing A.
Proof. Let f be a quasi g*p-open function.
Now, we have int(U)⊂U and g*p-int(U) is a
g*p-open set. Hence we obtain that f(g*pint(U)) ⊂f (U). As f(g*p-int(U)) is open,
f(g*p-int(U)) ⊂ int (f (U)).
Conversely, assume that U is a g*p-open set
in X. Then f (U) = f(g*pint(U)) ⊂ int (f (U))
but int (f (U)) ⊂f (U). Consequently f (U) =
int (f (U)) and hence f is quasi g*p-open.
The g*p-interior [respectively g*pclosure] of A is denoted by g*pint(A)
[respectively g*pcl(A)].
3.3 Lemma If a function f: X→Y is quasi
g*p-open, then g*p-int (f-1 (G)) ⊂f-1 (int (G))
for every subset G of Y.
2.6 Definition A subset S is called a g*pneighbourhood12 of a point x of X , if there
exists a g*p-open set U such that x ∈U ⊂ S.
Proof. Let G be any arbitrary subset of Y.
Then, g*p-int(f -1(G)) is a g*p- open set in X
and f is quasi g*p-open, then f(g*p-int(f-
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014)
1
(G))) ⊂ int(f(f-1(G))) ⊂ int(G). Thus g*pint(f -1(G)) ⊂ f-1(int(G))).
3.4 Theorem For a function f: X → Y, the
following are equivalent.
(i) f is quasi g*p-open.
(ii) For each subset U of X , f(g*p-int(U))
⊂int (f (U)).
(iii) For each x ∈X
and each g*pneighbourhood U of x in X, there exists
a neighbourhood V of f (∅) in Y such
that V ⊂f (U).
Proof (i) => (ii): It follows from Theorem
3.2.
(ii) => (iii) : Let x∈X and U be an arbitrary
g*p-neighbourhood of X in X. Then there
exist a g*p-open set V in X such that x∈V
⊂U. Then by (ii),we have f (V) = f(g*pint(V )) ⊂ int (f (V )) and hence f (V) = int
(f (V )).Therefore, it follows that f (V ) is
open in Y such that f (x)∈f (V ) ⊂ f (U).
(iii) => (i) : Let U be an arbitrary g*p-open
set in X. Then for each y ∈f (U), by (iii)
there exist a neighbourhood Vy of y in Y
such that Vy ⊂ f (U).
As Vy is a neighbourhood of y, there
exist an open set Wy in Y such that y ∈Wy
⊂Vy. Thus f(U) = U{Wy: y ∈ f(U)} which is
a open set in Y . This implies that f is quasi
g*p-open function.
3.5 Theorem A function f: X→Y is quasi
g*p-open iff for any subset B of Y and for
any g*p-closed set F of X , containing f-1
(B), there exist a closed set G of Y
containing B such that f-1 (G) ⊂F.
Proof. Suppose f is quasi g*p-open. Let B
⊂Y and F be a g*p-closed set of X
221
containing f-1 (B). Now, put G = Y-f (X - F).
It is clear that f-1 (B) ⊂F implies B ⊂G.
Since f is quasi g*p-open, we obtain G as a
closed set of Y. Moreover, we have f-1 (G)
⊂F.
Conversely, Let U be a g*p-open set
of X and put B = Y \f (U). Then X \ U is a
g*p-closed set in X containing f-1 (B). By
hypothesis, there exists a closed set F of Y
such that B ⊂ F and f-1 (F) ⊂X \U. Hence,
we obtain f (U) c Y \F. On the otherhand, it
follows that B ⊂F, Y \F ⊂ Y \B = f (U).
Thus, we obtain f (U) = Y \F which is open
and hence f is a quasi g*p-open function.
3.6 Theorem A function f: X→Y is quasi
g*p-open iff f-1 (cl (B)) ⊂g*pcl (f-1 (B)) for
every subset B of Y.
Proof. Suppose that f is quasi g*p-open. For
any subset B of Y, f-1 (B) ⊂g*p-cl (f1 (B)).
Therefore by Theorem 3-5., there exists a
closed set F in Y such that B ⊂ F and f-1 (F)
⊂ g*p-cl (f-1 (B)). Therefore, we obtain f-1 (cl
(B)) ⊂f-1 (F) ⊂g*p-cl (f-1 (B)).
Conversely, let B ⊂ Y and F be a g*p-closed
set of X containing f-1 (B).Put W = clY (B),
then we have B ⊂W and W is closed set and
f-1 (W) ⊂g*pcl (f-1 (B)) ⊂F. Then by
Theorem 3.5., f is quasi g*p-open.
3.7 Lemma Let f: X →Y and g :. Y → Z be
two functions and g°f: X →Z is quasi g*popen. If g is continuous injective, then f is
quasi g*p-open.
Proof. Let U be a g*p-open set in X, then
(g°f)(U) is open in Z. Since g o f is quasi
g*p-open. Again g is an injective continuous
function, f(U) = g-1(g o f(U))) is open in Y .
This shows that f is quasi g*p-open.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
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R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014)
ACKNOWLEDGMENTS
The author is grateful to the
University Grants Commission, for its
Financial support under UGC Minor
Research Project and also to PC Jabin
Science College, Hubli, Karnataka state,
India.
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