Download On Almost Regular Spaces

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1857 - 1862
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http://dx.doi.org/10.12988/ijma.2013.3493
On Almost Regular Spaces
K. Chandrasekhararao
Department of Mathematics
SASTRA University, Kumbakonam, INDIA
[email protected]
Copyright © 2013 K. Chandrasekhararao. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
The main focus of my paper is to introduce the properties of almost regular
spaces.
Keywords: Almost regular spaces, δ - continuous maps
Mathematics Subject Classification: 54D15
1. Introduction
In 1969 M.K.Singal and S.P Arya introduced the concepts of almost regular
spaces. For background material, papers [ 1 ] to [ 5 ] may be perused.
The purpose of this paper is to formulate and establish some results on almost
regular spaces.
2. Preliminaries
We require the following definitions
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K. Chandrasekhararao
Definition 2.1:
A subset A of a topological space ( X , τ ) is said to be regularly open if A = int
cl A . A subset A is said to be regularly closed if A = cl int A.
The family of all regular open sets constitutes a base for a topology τS on X .
This topology τS is known as the semi regularization of τ .
We note that τS ⊂ τ .
Definition 2.2 :
If τS = τ , then the space ( X , τ ) is said to be semi regular . Any regular space is
semi regular . Int cl A will be denoted by α A .
Definition 2.3 :
Let f : ( X , τ ) → ( Y , σ ) be a function . Then f is said to be δ - continuous if
for each x ∈ X and each open neighbourhood of f ( x ) there is an open neighbourhood
U of x such that
f(ασ)⊂αV.
Definition 2.4 :
f is said to be almost continuous if for each x ∈ X and each open neighbourhood
U of x such that f ( U ) ⊂ α V . f is δ - closed if f : ( X , τS ) → ( Y , σS ) is closed . f is
regularly open whenever U is regularly open .
3. Almost regular spaces
Definition 3.1 : ( X , τ ) is said to be almost regular if for each τ - regularly closed
subset A of X and each point x ∉ A there are disjoint τS - open sets U and V such that A
⊂ U and x ∈ V .
Theorem 3.1 : ( X , τ ) is said to be almost regular ⇔ ( X , τS ) is regular .
We require the following known results .
Lemma 3.1 : If A and B are disjoint open sets in ( X , τ ) , then τ α ( A ) and τ α ( B )
are disjoint open sets in ( X , τS ) containing A and B respectively .
Proof of the theorem :
Step 1 :
Suppose that ( X , τ ) is almost regular . Let C be a τS - closed subset of X and x
∉ C . Let be an indexed set .
On almost regular spaces
1859
We have C = ⋂∈ , where is a τ - regularly closed for each i .
Hence there is some i ∈ I such that x ∉ Cj .
But ( X , τ ) is almost regular .
Therefore , there exists disjoint τ - open sets U and V such that C ⊂ Cj ⊂ U and
x∈V.
By the lemma , there are disjoint τS - open sets and such that
C ⊂ U ⊂ and x ∈ V ⊂ .
Thus , ( X , τS ) is regular .
Step 2 :
Suppose that ( X , τS ) is regular .
Let C be a τ - regularly closed set and x ∉ C .
But ( X , τS ) is almost regular .
Consequently , there are disjoint τS - open sets U and V such that C ⊂ U and x ∈
V.
But τS ⊂ τ .
Hence U , V ∈ τ .
Hence ( X , τ ) is almost regular .
Theorem 3.2 :
( X , τ ) is almost regular ⇔ for each x ∈ x and each regularly open
neighbourhood U of x , there exists a regularly open set V such that x ∈ V ⊂ cl ( V ) ⊂
U.
Proof :
Step 1 :
Suppose that X is an almost regular space .
Let x ∈ X and let U be a regularly open set containing x .
But then X – U is a regularly closed set with x ∉ X – U .
But X is almost regular .
Hence there exists disjoint regularly open sets V and W such that x ∈ V and X –
U⊂W.
But then V ⊂ X – W and X – W ⊂ U .
Therefore , x ∈ V ⊂ cl ( V ) ⊂ cl ( X – W ) .
Accordingly , x ∈ V ⊂ cl ( V ) ⊂ U .
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K. Chandrasekhararao
Step 2 :
Suppose that for each x ∈ X and for each regularly open neighbourhood U of x ,
there is a regularly open set V such that x ∈ V ⊂ cl ( V ) ⊂ U .
Let F be a regularly closed set in X and let x ∉ F .
Then there exists a regularly open set V such that x ∈ V ⊂ cl ( V ) ⊂ X – F .
⇒ x ∈ V and F ⊂ X – c l ( V ) .
⇒ the sets V and X – c l ( V ) are regularly open sets with V ⊂
cl ( V ) .
⇒ V and X – V are disjoint .
⇒ X is almost regular .
Theorem 3.3 : Let X and Y be nonempty topological spaces . The product X × Y is
almost regular ⇔ both X and Y are almost regular .
Proof :
Step 1 :
Suppose that X × Y is almost regular .
Let x0 ∈ X .
Let U be any open neighbourhood of x0 .
Let y0 ∈ Y .
Then U × Y is a regularly open neighbourhood of ( x0 , y0 ) .
But X × Y is almost regular .
Hence there exists a regularly open set W in X × Y such that
( x0 , y0 )∈ W ⊂ cl ( W ) ⊂ U × Y .
Let V1 × V2 be s basic regularly open subset of X × Y such that
( x0 , y0 )∈ V1 × V2 ⊂ W.
⇒ ( x0 , y0 )∈ V1 × V2 ⊂ cl ( V1 × V2 ) ⊂ cl W ⊂ U × Y .
⇒ ( x0 , y0 )∈ V1 × V2 ⊂ cl ( V1 )× cl ( V2 ) ⊂ U × Y .
⇒ x0 ∈ V1 ⊂ cl ( V1 ) ⊂ U .
⇒ X is almost regular .
Similarly we can show that Y is almost regular .
Step 2 :
Suppose that X and Y are almost regular .
Let ( x0 , y0 )∈ X × Y .
Let W be a regularly open neighbourhood of ( x0 , y0 ) .
Then there exists a basic regularly open set U1 and V1 such that
x0 ∈ U1 ⊂ cl ( U1 ) ⊂ U and y0 ∈ V1 ⊂ cl ( V1 ) ⊂ V.
On almost regular spaces
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Hence ( x0 , y0 )∈ U1 × V1 ⊂ cl ( U1 ) × cl ( V1 ) ⊂ U × V ⊂ W .
⇒ ( x0 , y0 )∈ U1 × V2 ⊂ cl ( U1 × V2 ) ⊂ U × V ⊂ W .
⇒ X × Y is almost regular .
Lemma 3.2 : If A and B are subsets of ( X , τ ) , where A and B are regularly open and
A ⊂ B then cl ( A ) ⊂ cl ( int ( cl ( B ) ) .
Proof :
A⊂B.
⇒ A ⊂ cl ( B )
⇒ A ⊂ int ( cl ( B ) ) because A is regularly open .
⇒ cl ( A ) ⊂ cl ( int ( cl ( A ) ) .
Theorem 3.4 : Let f : ( X , τ ) → ( Y , σ ) be a regularly open , δ - continuous , α closed surjection . If ( X , τ ) is almost regular then ( Y , σ ) is almost regular .
Proof :
Let y ∈ Y .
Choose x ∈ Y such that f ( x ) = y .
Let U be a regularly open neighbourhood of y .
Since f isδ - continuous , it follows that f – 1 ( U ) is regularly open .
But X is almost regular .
Hence there exists a regularly open set Vin X such that x ∈ V ⊂ cl ( V ) ⊂ f – 1 (
U).
⇒ y ∈ f ( V ) ⊂ f ( cl ( V ) ) ⊂ U .
⇒ y ∈ f ( V ) ⊂ cl ( f ( V ) ) ⊂ U
…
…
(1)
Since f is regularly open , it follows that f ( v ) is regularly open .
Also f ( cl ( V ) ) is α - closed , because f is an α - closed function .
Take A = f ( V ) and B = f ( cl ( V ) ) in the lemma .
We have cl ( A ) ⊂ cl ( int ( cl ( B ) ) .
That is cl ( f ( V ) ) ⊂ cl ( int ( cl [ f cl V ] ) )
…
…
(2)
Substituting ( 2 ) in ( 1 ) , we obtain
y ∈ f ( V ) ⊂ cl ( int ( cl [ f cl V ] ) ) ⊂ U .
⇒ y ∈ f ( V ) ⊂ cl f ( V ) ⊂ U , because f ( cl V ) ⊂ cl f ( V ) .
⇒ y is almost regular
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K. Chandrasekhararao
References
1. K.Chandrasekhara Rao , Topology , Alpha Science , Oxford 2009 .
2. S.G.Crossley and S.K.Hilde - brand , Serni topological properties , Fund .
Math.74 ( 1972 ) , 233 - 254 .
3. T.Noiri , Semi continuous mappings , Athi Aecad. Naz. Lincei. Rend. Cl. Sci.
Fis. Mat. Natur., 54 ( 1973 ) , 210 - 214 .
4. M.K.Singal and S.P.Arya , On almost regular spaces , Glasnik. Matematicki,
Series III , 4 (1969) , 89 - 99 .
5. D.Sivaraj , Semi open set characterizations of almost regular spaces , Glasnik.
Matematicki , 21 ( 1986 ) , 437 - 440 .
Received: April 15, 2013