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ZANCO Journal of Pure and Applied Sciences
The official scientific journal of Salahaddin University-Erbil
ZJPAS (2017), 29 (1); 6-10
http://dx.doi.org/10.21271/ZJPAS.29.1.2
- Open Sets in Topological Spaces
Sami A. Hussein
Department of Mathematics, College of Basic Education, Salahaddin University,Erbil
Kurdistan Region, Iraq
ARTICLE INFO
ABSTRACT
Article History:
Received: 29/04/2016
Accepted: 19/09/2016
Published: 17/ 4/2017
Keywords:
The main aim of this paper is to introduce a new class of set called - open set.
some general properties of it are investigated. The relationships between this class
of set with other related classes of sets are studied.
supra-topology
nearly open sets
operators
*Corresponding Author:
Sami A.Hussein
[email protected]
1. INTRODUCTION
By a space
or
we mean a
topological space on which no separation
axiom is assumed unless explicitly stated. In
[Levine, N.1963] defined semi-open, which is
a subset of a space satisfies
,or
equivalently if there exists an open set such
hat
. [Njasted, O.,1965] defined a
subset
of a space
to be
open if
, which plays an important role
in the field of general topology. [Mashhour et
al., 1982] defined a subset of a space to be
pre-open if
, [ Abd-El-Monsef et
al.,1983] defined a subset of a space to be
-open if
, while in 1986
Andrijvic D. independently defined semipre-
open set which is equivalent to -open set. A
subset
of a space
to be b-open if
, The family of all semiopen(resp.,preopen,
open, -open and bopen)sets
are
denoted
by
[Khalaf, A.B. and Ameen, Z.A. ,2010]
introduced and investigated a new way to
define a new class of set called sc open set.
and used it in the way to study some properties
of continuity, Based on sc open set, pc-open,
bc-open and
-open sets were defined in
[Ameen, Z.A., 2011, Ibrahim, H.Z., 2013,
Mizyed, A Y., 2015]
As a generalization of sc-open,pc-open,bcopen and
-open sets. In this work, we
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Hussein, S. /ZJPAS: 2017, 29(1): 6-10
introduce and study a new class of set, called
- open sets, which is a subset of a space
for which if for each
, there
exists a closed set such that
.
exists
an
open
set
such
that
3.
2. Preliminaries
We recall the following definitions and results
which will be used in the sequel.
Definition 2.1:[ Mashhour A.S. et al.,1983]
Let
be a topological space and let be a
collection of subsets of
such that
.
Then
is called supra-topology associated
with , if satisfies the following conditions
1)
.
2) If
for each
arbitrary set, then
Definition 3.1: A subset
of a space
called
-open if for each
there exists a closed set such that
is
,
.
From now on, by
or simply
, we mean a family of all
-open
subsets of a space
is an
The relationship between -open set and some
other sets is given in the following results.
Definition 2.2: A subset
of a space
is
called sc-open [Khalaf, A.B. and Ameen, Z.A.
,2010] (resp., pc-open [Ameen, Z.A,2011], Bcopen [Ibrahim, H.Z.,2013], and
-open
[Mizyed, A Y.,2015]) if for each
Theorem 3.2: A subset of a space is open if and only if is -open and it is a union
of closed sets of .
there exists a closed set
, where
.
In this section we introduce our main
definition, which is
-open set, with some of
its properties.
such that
Definition 2.3: [Velicko N., 1968]A subset
of a space
is called -open (resp., - –
open) set, if for each
, there exists an
open( resp.,
–open) set
such that
. where
is a supra
topology.
Definition 2.4: [Ahmed N.K,1990] A space
is said to be
-regular if for every
and
every -closed set
such that
, there
exists disjoint open sets
and
such that
and
.
Theorem 2.5: [Ahmed N.K,1990]A space is
-regular if and only if for every
and
every -open set
such that
, there
Proof:
Assume that
. Then
, and for each
there exists a
closed set
such that
, thus
, which implies that
.
Conversely
suppose
that
and
, where is a closed
set which containing
, then for each
,implies that
, for some closed set
and
.
Hence
.
Remark 3.3: It is clear that from Definition
3.1,
-open set implies -open set, but the
converse may not be true in general, as shown
in the following example
Example
3.4:
then
, then
.
Let
,
if
we
,
take
and
but
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Hussein, S. /ZJPAS: 2017, 29(1): 6-10
Remark 3.5: From the fact that union of
arbitrary collection of closed sets may not be
closed in general, therefore -open sets may
not be closed sets, as shown in the following
example
Proof: Let be an open set in
Then
, and for any
is closed subset of , such that
which implies that
.
Example 3.6: Consider the usual space for R
and let
,
then
, but it is not closed set.
Proposition 3.12: If a space
every open set is -open set.
Proposition 3.7: The union of Arbitrary
collection of -open sets is -open set.
Proof: Let
be a collection of
open sets for each
, where
is an
arbitrary
set.
Then,
we
have
, and for any
there exists
such that
, for some
. Since
a
closed
, then
,
is regular, then
Proof: Let be an open subset of a space , in
the case where
, then
,
otherwise, let
, since is a regular space,
then there exists an open set
such that
. Thus we have that
and
. By Theorem
3.1,
.
Proposition 3.13: For a
, then there exists
set
such
that
Hence
,
.
Remark 3.8: The intersection of two -open
sets may not be
-open set in general, as
shown in the following example.
Example
.
3.9:
Let
.
If
,
take
we
then
and
,
Proof: From Remark 3.3, we have that
, it is suffices to show that
. Let
, if
,
then
, otherwise, let
, since
is
, then
is a closed set for each
,
and
.
Therefore
.
From Theorem 2.5. Remark 3.3, we have the
following result.
then
but
Proposition 3.14: For a
.
Note that from Proposition 3.7 and Remark 3.8,
the family of all
is a supra-topology
for X.
Proposition 3.15: Every
space is -open set.
Remark 3.10: and
are incomparable
in general. In Example 3.9,
, but
, and
but
.
Proof: Let be a -open subset of a space ,
if
, then
, otherwise, let
, then there exists an open set such that
.
Proposition 3.11: For a
open set is -open set.
So
follows that
every
-open subset of a
.
It
. From the fact
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Hussein, S. /ZJPAS: 2017, 29(1): 6-10
,
and
by
Theorem
3.2,
.
. The class of all
is denoted by
.
space
-sets of a
Remark 3.16: The converse of Proposition
3.15 may not be true in general, as shown in
the following example.
Proposition 4.2: For any space
Example 3.17: Let
, and be a co-finite
topology for . Since
is T1 –space, and
is an open set, then by Proposition
3.11,
, but is not -open set.
Proof: Clearly
are in
Let
Then
. This implies that
By
,
for
each
.
we
have
Thus
, therefore
4.3:
For
any
Proof: Let
we have
take
. Then for all
,
,
. In particular, if we
.
, then
Remark 4.4: The converse of Proposition 4.3
may not be true in general, let we take the
space as
in Example 3.9, we see that
but
.
Proposition 4.5: A subset of a space
closed in
if and only if
Proof: A subset
only if
,
open
If we take
B in
,
is
-
in .
is closed in
is an open in
if
of a space
is
, for each
space
.
closed for each
Definition 4.1: A subset
called
-set if
Then
.
Proposition
4. The Topology generated by
sets
for each
.
.
for all
Proposition
3.7,
.
,
The topological properties of the above
concepts are coincides with those supratopology.
. Let
(
Proof: Follows from Theorem 3.2 and fact that
union of finite closed sets is closed set.
1. A subset N of a space , is said to be
-neighborhood of x, if there exists an
-open set G such that
.
2. A point
, is called
-interior
point of , if there exists an -open set
G such that
.
3. A point
, is called -limit point
of , if for each
-open set G
containing x such that
.
4. A point
, is called
-closure
point of , if for each
-open set G
containing x such that
.
and
.
, where
Definition 3.19: For a subset , of a space
and
, then
is a
topology for .
Proposition 3.18: For a finite space , every
-open set is -clopen set.
For now, we define some topological concepts
in the following definition
,
,if and
, if and only
for
each
if
and
only
if
is - closed set in .
, then for any
- closed set
-closed for each
in .
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Hussein, S. /ZJPAS: 2017, 29(1): 6-10
REFERENCES
ABD
El-MONSEF M.E, El-DEEB S.N and
MAHMOUD R.A(1983), On β-Open Sets and βContinuous Mapping, Bull. Fac. Sci. Assiut Univ,
12 (1), (77-90).
AHMED N.K(1990), On some types of Separation
Axioms, M.Sc. Thesis, Salahaddin Univ.
AMEEN Z.A.(2011), pc-Open sets and pc-Continuity in
topological spaces, J. of Advanced Research in
pure Mathematics Vol.3,(1),(123-134).
ANDRIJEVIC D. (1986)., Semi-Preopen Sets, Mat.
Vesnik , 38, (24-32).
ANDRIJEVIC
D.
1996),
On
b-open
sets,
Mat.Vesnik,48,(59-64).
IBRAHIM H.Z.(2013), Bc-open sets in topological
spaces, J. of Advanced in pure Mathematics
Vol.3,(34-40).
KHALAF A.B. and AMEEN Z.A (2010), sc-open Sets
and sc-continuity in Topological Spaces, J. of
Advanced Research in Pure Mathematics, 2 (3), 87101.
LEVINE N. (1963). Semi-open sets and semi-continuity
in topological spaces. Amer. Math. Monthly,
70(1), 36-41.
MASHHOUR A.S., ABD EL-MONSEF M.E, and ElDEEB S.N (1982), On Precontinuous and Weak
precontinuous
mappings,
Proc.Math.Phys.Soc.Egypt 53, 47.
MASHHOUR A.S., ALLAM A.A., MAHMOUDa F.S
and
KHEDR
F.H.
(1983),
ON
SUPRATOPLOGICAL SPACES, Indian J.pure
appl. Math.14(4),502-510.
MIZYED A Y.(2015), Continuity and Separation
Axioms Based on βc-Open Sets, M.Sc. Thesis,
Islamic Univ. of Gaza.
Njasted O.(1965), On some classes of nearly open
sets,Pacific J.math.Vol.15 (3),(961-970).
VELICKO N. (1968), H-Closed Topological Spaces,
American Mathematical Society, 78 (2), 103-118.
.