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Transcript
```International Journal of Contemporary Mathematical Sciences
Vol. 10, 2015, no. 1, 1 - 11
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijcms.2015.4545
Nano PS -Open Sets and Nano PS -Continuity
Mohammed M. Khalaf and Kamal N. Nimer
Department of Mathematics
Faculty of Science, Al-Azahar University, Asuiat, Egypt
Current Address: Al-Zulfi College of Science, Majmaah University
Saudi Arabia
Copyright © 2014 Mohammed M. Khalaf and Kamal N. Nimer. This is an open access article
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The purpose of this paper is to introduce a new class of sets called Nano ππ -open
sets in Nano-topological spaces. This class of sets lies strictly between the classes
of Nano πΏ- open and Nano -preopen sets. By using this sets we defined and study
the concept of Nano ππ -continuous function
Keywords: Nano topological space, Nano PS -open, Nano PS -continuous function
1. Introduction
Lellis Thivagar [1] introduced a nano topological space with respect to a
subset X of an universe which is defined in terms of lower and upper
approximations of X. The elements of a nano topological space are called the
nano-open sets. He has also studied nano closure and nano interior of a set. Njastad
[5], Levine [2] and Mashhour [3] respectively introduced the notions of - open,
semi-open and pre-open sets. Since then these concepts have been widely
investigated. It was made clear that each -open set is semi-open and pre-open but
the converse of each is not true. Njastad has shown that the family of - open sets is a
topology on X satisfying. The families SO (π, π) of all semiβopen sets and
PO(π, π) of all preopen sets in (π, π) are not topologies. It was proved that both
SO(π, π) and PO(π, π) are closed under arbitrary unions but not under finite
intersection.
In this paper we introduce a new class of sets called Nano PS -open sets in Nanotopological spaces and study the concept of Nano PS -continuous function.
2
Mohammed M. Khalaf and Kamal N. Nimer
Throughout this paper, a space ππ (π) is a topology on U called the nano
topology on with respect to X . We recall the following definitions, notions and
characterizations.
Definition 1.1 [1] Let π be the universe, π be an equivalence relation on π and
ππ (π) = {π, β, πΏπ (π), ππ (π), π΅π (π)} where X ο U . If ππ (π) satisfies the
following axioms:
(i) U , ο¦ οο΄ R ( X )
(ii) The union of the elements of any subcollection of ππ (π) is in ππ (π)
(iii) The intersection of the elements of any finite subcollection of ππ (π) is in
ππ (π)
Then ππ (π) is a topology on U called the nano topology on with respect to X .
We call as the NANO TOP space The elements of ππ (π) are called as nano-open
sets.
Definition 1.2 [1] If (π, ππ (π)) is NANO TOP space with respect to X where X
οο  U and if A οο  U , then the nano interior of A is defined as the union of all
nano-open subsets of A and it is denoted by N int(A) . That is, N int(A) is the
largest nano-open subset of A . The nano closure of A is defined as the
intersection of all nano closed sets containing A and it is denoted by Ncl (A) .
That is, Ncl (A) is the smallest nano closed set containing A .
Definition 1.3 [1] A NANO TOP space (π, ππ (π)) is said to be extremally
disconnected, if the nano closure of each nano-open set is nano-open.
Definition 1.4 Let ππ (π) be a NANO TOP on U with respect to X . A subset
A of U is said to be nanosemi-open [2] (by short N ππ -open) (resp.
nanopreopen [3] (by short N π-open) , nano ο‘ -open [4] (by short N πΌ-open),
nano- ο’ -open (by short N π½-open) or nanosemi-preopen (by short N ππ-open),
nano- b -open(by short N π-open) ) if A ο cl (int ( A))
(resp.
A ο int (cl ( A)), A ο int (cl (int ( A))), A ο cl (int (cl ( A))), A ο cl (int ( A)) ο int (cl ( A))
).
The family of all nano semi-open (resp. nanopreopen, nano- ο‘ -open, nano- ο’
-open, nano- b -open) sets in X is denoted by NSO(U , X ) (resp. NPO(U , X ) ,
Nο‘ (U , X ) , Nο’ (U , X ) , NbO(U , X )
The intersection of all NPC (U , X ) (resp. NSC (U , X ) and Nο‘C (U , X ) ) sets of U
containing a subset A is called the NPclosure(U , X ) (resp. NSclosure(U , X ) and
Nο‘ closure(U , X ) ) of A . The union of all NPO(U , X ) (resp. NSO(U , X ) ) sets of
U contained in A is called the NPinterior (U , X ) (resp. NSinterior (U , X ) ) of A
Nano ππ -open sets and nano ππ -continuity
3
Definition 1.5 Let ππ (π) be a NANO TOP on U with respect to X . A subset
A of U is said to be
1- A subset A of U is said to be nanoregular-open (by short N π -open) if
A ο½ N (int (cl ( A)))
2- Nano πΏ -open (by short Nο€O(U , X ) ) if for each x ο A , there exists an
Nro(U , X ) G such that x ο G ο N (int (cl (G))) ο A
3- Nano π -open (by short Nο±O(U , X ) ) if for each x ο A , there exists an
NO(U , X ) G such that x ο G ο N (cl (G))) ο A
The intersection of all N πΏ-closed sets of U containing A is called the Nano πΏ
-closure (by short ππππΏ (π΄)). The union of all N πΏ-open sets of U containing A
is called the Nano πΏ -interior (by short ππππ‘πΏ (π΄)).
4- is Nano-semi dense set (by short NS dense(U , X ) ) if N (cl (int (clA))) ο½ A
Definition 1.6 [1] Let ππ (π) be a NANO TOP on U with respect to X . A
subset A of U is said to be nano- ο€ -semi-open (resp. nano- ο€ -preopen ) if
A ο cl ( Intο€ ( A)) (resp. A ο int (clο€ ( A)) ).
Definition 1.7 Let ππ (π) be a NANO TOP on U with respect to X . A subset
A of U is said to be Nano ππ -open (by N ππ -open ) if for each
x ο A ο NPO(U , X ) , there exists a Nano semi-closed (by short NSC (U , X ) ) set
F such that x ο F ο A . The family of all N ππ -open subsets of ππ (π) is
denoted by NPS O(U , X )
Proposition 1.1 A subset A of U is said to be N ππ -open if and only if A is N
ππ-open and union of
NSC (U , X ) sets
Proof. Obvious
Proposition 1.3 If A and B are N ππ -open subsets of is NANO TOP space
(π, ππ (π)) and if the family of all N ππ-open sets in U forms a NANO TOP
ππ (π) of U , then A β© B is N ππ -open set and hence the family of N ππ -open
sets forms a NANO TOP ππ (π) of U
Definition 1.8 A space (π, ππ (π)) is semi ο­ T1 if for each pair of distinct points
x, y οU , there exists a pair of NSO(U , X ) sets containing x but not y and the
other containing y but not x
Theorem 1.1 A space (π, ππ (π))
singleton set {x} is NSC (U , X ) set
is semi ο­ T1 if for any point x οU , the
4
Mohammed M. Khalaf and Kamal N. Nimer
Theorem 1.2 If a space (π, ππ (π)) is semi ο­ T1 , then
NPSO(U , X ) ο½ NPO(U , X )
Proof. Let A ο U and Aο NPO(U , X ) . If A ο½ ο¦ , then Aο NPO(U , X ) If
A οΉ ο¦ , then for each x ο A . Since a space (U ,ο΄ R ( X )) is semi ο­ T1 , then by
Theorem 1.1, {x} is NSC (U , X ) set and hence x ο{x} ο A . Therefore
Aο NPSO(U , X ) . Hence NPO(U , X ) ο NPSO(U , X ) ). But
NPSO(U , X ) ο NPO(U , X ) in general. Therefore, NPSO(U , X ) ο½ NPO(U , X )
Definition 1.9 A space (U ,ο΄ R ( X ) is Nano s-regular (by short N π π(π, π)) if for
each x οU and each
NO(U , X ) set G containing x , there exists a NSO(U , X ) set H
such that x ο H ο NSclosure( H ) ο G
Lemma 1.1 For any subset A of a space (π, ππ (π)), then Aο NPO(U , X ) if
and only if
NSclosure(A) ο½ N (int(cl ( A)))
Theorem 1.3 For any subset A of a space (π, ππ (π)). If A ο Nο€O(U , X ) , then
Aο NPSO(U , X )
Proof. Let A ο Nο€O(U , X ) . If A ο½ ο¦ , then Aο NPSO(U , X ) . If A οΉ ο¦ , then
for each x ο A there exists an NO(U , X ) set B such that
x ο B ο N (cl (int ( B))) ο A . Since B is NO(U , X ) and hence B ο NPO(U , X ) .
By Lemma 1.1 NSclosure((B)) ο½ N (int(cl ( B))) . Then x ο B ο NSclosure(( B)) ο A
implies that x ο NSclosure(( B)) ο A and NSclosure(( B)) ο NSC (U , X ) Since
A ο Nο€O(U , X ) and
Nο€O(U , X ) ο NPO(U , X ) in general, Aο NPO(U , X ) Therefore
Aο NPSO(U , X )
Remark 1.1
1. Each N (cl ( NO(U , X )) and hence each Nro(U , X ) set is N ππ -open
2. Each Nο±O(U , X ) set is N ππ -open
Lemma 1.2 A space (π, ππ (π)) is hyper connected if and only if Nro(U , X )
ο½ {U , ο¦}
Theorem 2.4 A space (π, ππ (π) is hyper connected if and only if
NPSO(U , X ) ο½ {U , ο¦}
Nano ππ -open sets and nano ππ -continuity
5
Proof. Necessity. Suppose, if possible, that a subset Aο NPSO(U , X ) If A ο½ ο¦ ,
then Aο NPSO(U , X ) . If A οΉ ο¦ , then for each x ο A there exists an
F ο NSC (U , X ) such that x ο F ο A . Since F ο NSC (U , X ) , then
NSclosure(int (clF )) ο F . Since (U ,ο΄ R ( X ) is hyper connected space, then for every
nonempty NPO(U , X ) ο U is NS dense(U , X ) , so N (cl (int (clF ))) ο½ U ο F , this
implies that U ο½ F . Since F ο½ U ο A , then U ο½ A Therefore, A must be equal to
U , which completes the proof.
Sufficiency. Suppose that NPSO(U , X ) ο½ {U , ο¦} . Since Nro(U , X ) subset of
NPSO(U , X ) in general, then Nro(U , X ) ο½ {U , ο¦} . By Lemma 1.2, we have
(U ,ο΄ R ( X ) is hyper connected.
Definition 1.10 A space (π, ππ (π)) is called locally indiscrete if every
NO(U , X ) is NC (U , X )
Lemma 1.3 If U is locally indiscrete space, then each NSO(U , X ) of U is
NC (U , X ) and hence each NSC (U , X ) of U is NO(U , X )
Theorem 2.5 If a nano topological space (π, ππ (π)) is called locally indiscrete,
then NPSO(U , X ) ο½ ο΄ R ( X )
(π, ππ (π)) is locally indiscrete space and let A ο U such that
A οο΄ R (U ) . Since every NO(U , X ) is NC (U , X ) , then N (int (clA)) ο½ A this
implies that Aο NrO(U , X ) . Hence by Remark 1.1 (1), Aο NPSO(U , X ) . Thus,
ο΄ R (U ) ο NPSO(U , X ) .
On the other hand, let B ο NPSO(U , X ) , then B ο NPO(U , X ) and for each
x ο B , there exists F ο NSC (U , X ) such that x ο F ο B . By Lemma 1.3, F is
NO(U , X ) . It follows that B οο΄ R (U ) . Hence NPSO(U , X ) οο΄ R ( X ) . Therefore,
Proof Let
NPSO(U , X ) ο½ ο΄ R ( X )
The following diagram show relations among NPSO(U , X ) , NPO(U , X ) ,
NrO(U , X ) , Nο€O(U , X ) , ππ (π)and Nο‘ (U , X )
NrO(U , X )
ο
NPSO(U , X )
ο
Nο€O(U , X )
ο
NPO(U , X )
ο
Nο‘ (U , X )
ο
ππ (π)
6
Mohammed M. Khalaf and Kamal N. Nimer
Remark 1.2 In above Diagram, we have the following notes
1. ππ (π) is not comparable with NPSO(U , X )
2. Nο‘ (U , X ) is not comparable with NPSO(U , X )
Definition 1.11 A subset B of a space (π, ππ (π)) is called N ππ -closed if
U B is N ππ -open set.
The family of all N ππ -closed subsets of a NANO TOP space (π, ππ (π)is
denoted by NPS C (U , X )
Proposition 1.2 A subset B of a space (π, ππ (π))
and only B is an intersection of NSO(U , X )
is called NPC (U , X ) if
Proof. Obvious
Corollary 2.1 For any A of a space (π, ππ (π)) . The following conditions are
equivalent:
1.
2.
3.
4.
2.
A
A
A
A
is
is
is
is
N closure( NO(U , X ))
N ππ -open and NC (U , X )
Nο‘ (U , X ) and NC (U , X )
NPO(U , X ) and NC (U , X )
N π·πΊ -Continuous Functions
Definition 2.1 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
N π·πΊ -continuous at a point x οU if for each x οU and each open set B of a
space (π, π β π (π))
Containing f (x) , there exists a NPS O(U , X ) A of a space (π, ππ (π))
containing x such that f ( A) ο B . If f is N π·πΊ -continuous at every point of U ,
then it is called N π·πΊ -continuous
Definition 2.2 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
N -continuous if the inverse image of each NO(U , X ) subset of a space
(π, π β π (π)) is NO(U , X ) in (π, ππ (π))
Nano ππ -open sets and nano ππ -continuity
7
Definition 2.3 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
NP -continuous ( resp. Nο‘ -continuous, N -perfectly continuous, N -complete
continuous, N super-continuous) if the inverse image of each NO(U , X ) subset
of A of a space (π, π β π (π)) is NPO(U , X ) ( resp. Nο‘ (U , X ) ,
N closureO(U , X ) , NRO(U , X ) , Nο€O(U , X ) )
in (π, ππ (π))
Definition 2.4 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
N -Almost pre continuous ( resp. N -Almost continuous,) if the inverse image of
each NRO(U , X ) A of a space (π, π β π (π)) is NPO(U , X ) ( resp. NO(U , X )
)in (π, ππ (π))
Definition 2.5 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
Nο€ - continuous ( resp. Nο± -Almost continuous ) if for each x οU and each
NO(U , X ) A of a space (π, π β π (π)) containing f (x) , there exists an
NO(U , X ) B of a space (π, ππ (π)) containing x such that
f ( N (int(clB ))) ο N (int(clA)) ( resp. f ( N (cl ( B)) ο N (cl ( A)) )
Definition 2.6 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively. is called
N - contra continuous ( resp. N -contra semi continuous ) if the inverse image of
every NO(U , X ) subset of (π, π β π (π)) is NC (U , X ) ( resp. NSC (U , X ) ) of
a space (π, ππ (π))
Lemma 2.1 Every Ncomplete -continuous and hence every N super-continuous
function is N π·πΊ -continuous and every N π·πΊ -continuous function NP
-continuous.
Proof. Follows directly from their definitions.
The following diagram show relations among N -complete continuous, N
super-continuous, N π·πΊ -continuous, N -continuous, Nο‘ -continuous and NP
-continuous
8
Mohammed M. Khalaf and Kamal N. Nimer
N -complete
continuous
β
N super-continuous,
β
β
N -continuous
β
-continuous
Nο‘
N
π·πΊ -continuous
β
NP -continuous
Theorem 2.1 For a function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X
and Y respectively, the following statements are equivalent:
1. π is N π·πΊ -continuous
2. π β1 (π΄) is N π·πΊ -open set in (π, ππ (π)) of each NO(U , X ) A in
(π, π β π (π))
3. π β1 (πΉ) is N π·πΊ -closed set in (π, ππ (π)) of each NC (U , X ) F in
(π, π β π (π))
4. For each NO(U , X ) A in (π, ππ (π)), π( N π·πΊ (πππ(π΄))) β π(ππ(π(π΄))
5. For each NC (U , X ) F in (π, π β π (π)), ππ·πΊ (πππ(π β1 (πΉ))) β π β1 (πππ(πΉ)
6. For each NO(U , X ) B in (π, π β π (π)), π β1 (ππππ‘(π΅)) β πππ (ππππ‘(π β1 π΅))
Proof. Straight forward.
Theorem 2.2 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively, is
N π·πΊ -continuous if and only if π is NP -continuous and for each x οU and
each NO(U , X ) B of V containing f (x) , there exists a NSC (U , X ) F in
U containing x such that f ( F ) ο B
Proof. Necessity. Let x οU and let V be any NO(U , X ) B containing f (x) .
Since π is
N π·πΊ -continuous, there exists a N ππ -open set A of U containing x such that
f ( A) ο B . Since A is N ππ -open set. Then for each x οU , there exists a
NSC (U , X ) F in U such that x ο F ο U Therefore, we have f ( F ) ο B . And
also since π is N π·πΊ -continuous,. Then π is NP -continuous
Sufficiency. let V be any NO(U , X ) V . We have to show that π β1 (π΅) is N
ππ -open set in U . Since π is NP -continuous , thenπ β1 (π΅) is NPO(U , X ) in
U . Let π₯ β π β1 (π΅). Then f ( x) ο B By hypothesis, there exists a NSC (U , X )
Nano ππ -open sets and nano ππ -continuity
9
F of U containing x such that f ( F ) ο B . Which implies that x ο F ο f ο­1 (B)
Therefore, π β1 (π΅) is N ππ -open set in U . Hence by Theorem 2.1, π is N
π·πΊ -continuous
Theorem 2.3 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively, is
N π·πΊ -continuous and NO(U , X ) function and B is NPO(U , X ) in V , π β1 (π΅)
is NPO(U , X ) in U
Proof. Let B be a NPO (U , X ) in V , then there exists an N -open set G in V
ο­1
ο­1
ο­1
, such that B ο G ο N (cl ( B)) . Then f ( B) ο f (G) ο f ( N ((cl ( B))) . Since
f is N -continuous function, f ο­1 (G) is NO(U , X ) in U and since f is an
NO(U , X ) function, then f ο­1 (cl ( B)) ο N (cl ( f ο­1 (G)) ). Hence we obtain that
f ο­1 ( B) ο f ο­1 (G) ο N ο­ cl ( f ο­1 (( B))) ). Therefore, π β1 (π΅) is NPO(U , X ) in U
Theorem 2.4 If π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y
respectively, is N -continuous and
NO(U , X ) function and F is a NSC (U , X ) of V , then f ο­1 ( F ) is a
NSC (U , X ) of U
Proof. Let be a NSC (U , X ) of V , then there exists a NC (U , X )
ο­1
ο­1
E in V
ο­1
such that N (int(E) ο F ο E . Then f ( N (int(E) ο f ( F ) ο f ( E) ). Since
f is N -continuous function, then f ο­1 ( F ) is a NC (U , X ) in U and
since f
is an NO(U , X )
ο­1
ο­1
function, then N (int( f ( E)) ο f ( N (int(F )) ).
ο­1
ο­1
ο­1
Hence we obtain that N ο­ int( f ( E )) ο f ( F ) ο f ( E) ). Therefore,
f ο­1 ( F ) is NSC (U , X ) of U
Theorem 2.4 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and
Y respectively, is a
N -continuous and NO(U , X ) function and B is a N ππ -open set of V , then
π β1 (π΅) is
N ππ -open set in U
Proof. Let B is a N ππ -open set of V , then B is NPO(U , X ) in V and
B ο½ οFi is where Fi is
NSC (U , X ) of V for each i . Then π β1 (π΅) =
π β1 (βͺ πΉπ ) =βͺ π β1 (πΉπ ), where Fi is NSC (U , X ) of V for each i . Since f is
a N -continuous and NO(U , X ) function Then by Theorem 2.3 π β1 (π΅) is
10
Mohammed M. Khalaf and Kamal N. Nimer
NPO(U , X ) in U and by Theorem 2.4 π β1 (βͺ πΉπ ) NSC (U , X ) of U for
each i . Hence by Proposition 1.1, π β1 (π΅) is N ππ -open set in U
Corollary 2.1 If π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y
respectively, is N -continuous and
NO(U , X ) function and F is a N ππ -closed set of V , then π β1 (πΉ) is N
ππ -closed set in U
Theorem 2.5 The following statements are equivalent for a function
π: (π, ππ (π)) β (π, π β π (π)). With respect to X and Y respectively,
1. f
2. f
3. f
4. f
is
is
is
is
N -perfectly continuous
N π·πΊ -continuous and N - contra continuous
Nο‘ -continuous and N - contra continuous
NP -continuous and N - contra continuous
Proof. This is an immediate consequence of Corollary 2.1
Theorem 2.6 Let π: (π, ππ (π)) β (π, π β π (π)). with respect to X and Y
respectively, be a function. Let πβπ β
be any basis for (π, π β π (π)) . Then f is N π·πΊ -continuous if and only if for each
πβ β π β1 (πβ β ) is
π
π
N π·πΊ -open subsets of U
Proof. Necessity. Suppose that f is N π·πΊ -continuous. Then since each πβπ β N
π·πΊ -open subsets of V and f is N π·πΊ -continuous. Therefore, by Theorem 2.1,
π β1 (πβπβ ) is N π·πΊ -open subsets of U
Sufficiency. Let B is a N ππ -open set of V . Then B = {βͺ (πβ β ) : π β πΌ}
ππ
where every πβ β is a member of πβπ β and I is a suitable index set. It follows
ππ
that
f ο­1 ( B) ο½ π β1 {βͺ (πβ
ππβ
) : π β πΌ} =βͺ π β1 {(πβ β ) : π β πΌ} }. Since
ππ
ο­1
π β1 {(πβ β )} is N π·πΊ -open subsets of U for each β πΌ . Hence f ( B) is
ππ
Nano ππ -open sets and nano ππ -continuity
11
the union of a family of N π·πΊ -open subsets of U and hence is N π·πΊ -open subsets
of U Therefore, by Theorem 2.1, f is N π·πΊ -continuous
References
[1] M. Lellis Thivagar and Carmel Richard, Note on nano topological spaces,
Communicated.
[2] N. Levine, Semiβopen sets and semicontinuity in topological spaces, Amer.
Math. Monthly 70 (1963), 36 β 41. http://dx.doi.org/10.2307/2312781
[3] A. S. Mashhour, M. E. Abd El-Monsef and S.N. El-Deeb, On pre-topological
spaces, Bull.Math. de la Soc. R. S. de Roumanie 28 (76) (1984), 39 β 45.
[4] Miguel Caldas, A note on some applications of -open sets, IJMMS, 2 (2003),
125-130 O. Njastad, on some classes of nearly open sets, Pacific J. Math. 15 (1965),
961 β 970.
[5] Z. Pawlak, Rough sets, International journal of computer and Information
Sciences, 11 (1982), 341 - 356. http://dx.doi.org/10.1007/bf01001956
[6] I. L. Reilly and M. K. Vamanamurthy, On-sets in topological spaces, Tamkang
J. Math., 16 (1985), 7 - 11.
Received: May 15, 2014; Published: January 3, 2015
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