Download on α-i alexandroff spaces

Asian Journal of Current Engineering and Maths 2: 3 May – June (2013) 205 - 207.
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ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
ON Α-I ALEXANDROFF SPACES
I. Arockiarani, A. A. Nithya
Department of Mathematics with Computer Applications, Nirmala College for Women, Coimbatore, India
ARTICLE INFO
ABSTRACT
Corresponding Author
A. A. Nithya
Department of Mathematics with
Computer Applications,
Nirmala College for Women,
Coimbatore, India
Key Words: P -space, P 0 -space,
The objective of this paper is to explore the idea of ideal in α-IAlexandroff spaces. Further we have proved some equivalent conditions
and arrived a condition in which τ and τα coincide.
Alexandroff-space, α-I-open, αI- Alexandroff-space, semiAlexandroff-space. base B(I,τ )=
{U \G : U ∈ τ and G ∈ I}.
Additionally, cl*(A) =A∪A* defines
∗ 2000 Math. Subject Classification — Primary: 54G05, 54G10,
Secondary: 54H05, 54G99.
INTRODUCTION
Intensive development and research have been
conducted in the field of Alexandroff spaces in topology
and also in digital topology. This research area which is
currently gaining much interest, which paved the way for
digital concept. A subcollection I is said to be ideal if it
satisfies 1) If A ∈ I and B
⊂ A then B ∈ I. 2) If A,B ∈ I then A ∪ B ∈ I. Then the
triple (X,τ ,I) is said to be ideal topological space. In an
ideal topological space for a subset A of X, A*(I)={x ∈ X
: U ∩ A ∉ I for each neighbourhood U of x } is called the
local function of A with respect to I and τ . For every
ideal topological space (X,τ ,I), there exists a topology τ *
(I), finer than τ , generated by the a Kuratowski closure
operator for τ *(I).
A topological space (X, τ ) is said to be locally finite if its
each element x of X is contained in a finite open set and
a finite closed set.
It is clear that all finite topological spaces are locally
finite and all locally finite topological spaces are
Alexandroff.
In this paper, we consider a weaker form of Alexandroff
spaces called α-I- Alexandroff.
Definition 1.1. Let A be a subset of a space (X,τ ), then
A is said to be
[a] pre-open[12] if A ⊂ int(cl(A)), [b] semi-open[11] if A
⊂cl(int(A)), [c] β-open[1] if A ⊂ cl(int(cl(A))). [d] α-open
if A ⊆ int(cl(int(A))).
We denote the family of all pre-open sets, semi-open sets,
β-open sets and α-open sets as PO(X,τ ),SO(X,τ ) βO(X,τ )
and αO(X, τ ).
Definition 1.2 A subset S of a space (X, τ ,I) is said to
be
1. I-open if S ⊆ int (S*),
2. pre-I-open if S⊆ int(cl*(S))
3. semi-I-open if S⊆ cl* (int (S)).
©2013, AJCEM, All Right Reserved.
4. α-I-open if S ⊆ int (cl*(int(S))).
Complements of α-I-open sets are called α-I-closed.
The union of all α-I-open sets contained in A is said be
α-I-int(A). The intersection of all α-I-closed sets
containing A is said to be α-I-cl(A).
A topological space (X, τ ) is a P ‘ -space if and only if
arbitrary countable intersections of open sets are semiopen.
A topological space (X, τ ) is called open hereditary
irresolvable [4] if each open subset of X is irresolvable
and quasi-maximal [4] if every dense set with non-empty
interior has dense interior. A subset A ⊆ X is said to be
simply- open [18] if A = U ∪ N , where U is open and N is
nowhere dense (a set S is nowhere dense if int(cl(A))= φ.
In [4], Chattopadhyay and Roy called A a δ-set if
intcl(A) ⊆ cl(intA). It is easily observed that a set A is
simply-open if and only if it is a δ-set.
A topological space (X, τ ) is called door space if each A
⊆ X is either open or closed.
α-I-Alexandroff
Definition 2.1 A topological space (X, τ, I) is called α-IAlexandorff if arbitrary intersection of open sets is α-Iopen.
Remark 2.2 An α-I Alexandorff space need not
necessarily satisfy the con- dition ”Arbitrary
intersections of α-I-open sets are α-I-open“.
Lemma 2.3 If a space (X, τ, I) contains a point p such
that {p} is open and dense, then (X, τ ,I) is α-IAlexandorff.
Proof. Let {Oi : i ∈ I} be a collection of open sets and O be
its intersection.
Suppose that O is not α-I-open, i.e. O ⊄ int(cl ∗ (int(O))).
Then there exists x ∈ O and an open set V containing x
with V ∩ int O = ∅. Since {p} is open and dense, then
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we have p ∈ V and p ∈ Oi for each i ∈ I, hence also
p ∈ int O, w h i c h i s a contradiction. Thus O is α-I-open.
Theorem 2.4 Let (X, τ ,I) be an ideal topological space
then every Alexandorff space is α-I-Alexandorff space.
Proof. Let Oi be the arbitary collection of open sets,
since the space is Alexandorff therefore its intersection is
open set. We know that every open set is α-I-open. Thus
intersection of arbitary open sets is α-I-open. Hence the
proof.
Remark 2.5 Every α-I-Alexandorff is not always
Alexandorff is shown in the following example.
Example 2.6 Let X=R be the real line with the topology
τ = {∅, X, {0}} ∪{G ⊆ X : 0 ∈ G and X \G is finite},
I={φ}. Clearly {0} is open and dense in (X, τ, I) and
so (X, τ, I ) is α-I-Alexandorff by Lemma.2.3. On the
other hand, the intersection of all open sets X \ {x},
where x is irrational, is not open. Thus (X, τ, I ) is not
Alexandorff.
Theorem 2.7 For an ideal topological space (X, τ .I) the
following conditions are equivalent:
(1) X is T1 and α-I-Alexandorff. (2) X is discrete.
Proof. (1) ⇒ (2) For each x ∈ X and each y = x, there
exists an open set
Ux containing x such that y ∉ Ux (X is T1 ). Since X is αI-Alexandorff and since {x} = ∩y=x Ux , then {x} is αI-open and hence open, since a singleton is α-I-open if
and only if it is open. Thus X is discrete.
(2) ⇒ (1) Let X be discrete space, then every singleton
set is open. Therefore
for any two points x and y we can find two open sets
such that x ∈ U and y 6∈ U similarly y ∈ V and x ∉ V .
Hence it is T1 space. Also, since the underlying space
is discrete arbitrary intersection of open sets is α-Iopen. Hence the proof.
Lemma 2.8 [10]For a subset A of a space X the
following conditions are equivalent:
(1) A is a simply-open set.
(2) A is the intersection of a α-I-open and a α-Iclosed set (ie) A∈LC(X, τ α ).
Lemma 2.9 [4] For a topological space X the following
conditions are equivalent:
(1) Every subset of X is simply-open.
(2) X is open hereditary irresolvable and quasi-maximal.
Theorem 2.10 Let (X, τ ) be T 1
and α-I-Alexandorff.
Then X is open hereditary irresolvable and quasimaximal.
Proof. According to Lemma 2.9 and 2.8, we need to
show that every subset of X is simply-open. Let A ⊆ X. Set
A = A1 ∪ A2 (with A1 ∩ A2 = ∅), where each point of A1
is closed in X and each point of A2 is open in X (this is
possible, since X is T 1 ). Since X is α-I-Alexandorff, then
A1 is α-I-closed and moreover A2 is α-I-open. Using
Lemma 2.8, each subset of X is the complement of a
simply-open set, i.e. each subset of X is simply-open.
Theorem 2.11 Every α-I-Alexandorff space is a P ‘ -
space.
Proof. Let us take a countable collection of open set in X,
Since it is α-I Alexandorff every arbitrary intersection of
open sets is α-I-open and thus it is semi open which
implies X is P ‘ space. Hence the proof.
Remark 2.12 However the converse need not be true
is shown by the fol- lowing example.
Example 2.13 Let X be the real line where the nontrivial open sets are all sets containing the zero point
and having countable complements. Clearly this is a P space and hence a P 0 -space. The intersection of all
open sets of the form X \ {x}, for x = 0, is not α-Iopen. This shows that X is not α-I-Alexandorff.
Theorem 2.14 Let X be a door space then the
following conditions are equivalent:
(1) X is Alexandorff.
(2) X is α-I Alexandorff.
Proof. (1)⇒(2) It is straightforward. (2)⇒(1) Let us
consider a collection of open sets whose intersection is α-I
open. Since X is a door space all subsets of X will be either
open set or closed set. In this case αIO(X) = τ .
Therefore in this case every α-I open set is open set.
Thus it is Alexandorff space.
Theorem 2.15 Let f : (X, τ, I ) → (Y, σ, J ) be a surjective
continuous α-I- open function. If X is Alexandorff, then Y
is α-I-Alexandorff.
Proof. To prove Y is α-I-Alexandorff, let {Oi |i ∈ I} be
the arbitrary
collection of open sets in Y. Since f is continuous, we
have f−1 (Oi ) is open X. Since X is Alexandorff
intersection of this collection is open set G, then using f
is surjective and α-I open we get f(G) is α-I-open in Y.
f(G)=f ∩ f −1 (Oi ).
=(∩(f f −1 (Oi |i ∈ I))
=∩(Oi |i ∈ I) in Y. Thus Y is α-I- Alexandorff.
Definition 2.16 Let (X, τ ,I) be a ideal topological space,
then if cl∗ (int(A))
= int(A), then it is said to be ic*-set.
Definition 2.17 A function f : (X, τ, I) → (Y, σ, J ) is
said to be ic*- continuous if for every open set in Y,
then its inverse image is an ic* set.
Theorem 2.18 A function f : (X, τ, I) → (Y, σ, J ) is
continuous if and only if it is α-I-continuous and ic*continuous.
proof. If f is continuous then it is always α-I-continuous
and ic*-continuous. Conversely, Since f is α-I-continuous
and ic*-continuous, for every open set U
in Y f −1 (U ) is α-I-open and ic*-set. Therefore f−1 (U ) ⊆
int(cl∗(int(f −1 (U ))) and cl ∗ (int(f −1 (U ))) = int(f
−1 (U )). Thus we get f −1 (U ) ⊆
int(f −1 (U )) and always int(f −1 (U )) ⊆ f −1 (U ).Hence f
−1 (U ) = int(f −1 (U )), (i.e)f −1 (U ) is open in X.
Theorem 2.19 Let X be α-I-Alexandorff and let f : (X, τ,
I) → (Y, σ, J ) be
α-I-open function, continuous and onto. Then Y is α-IAlexandorff.
Proof. To prove Y is α-I-Alexandorff, let {Oi |i ∈ I } be
the arbitrary collection of open sets in Y. Since f is
continuous, we have f−1 (Oi ) is open X. Since X is α-I
Alexandorff, intersection of this collection is α-I- open
set G, then using f is surjective and α-I-open function,
we get f(G) is α-I-open in Y.
f(G)=f ∩ f −1 (Oi ).
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=(∩(f f −1 (Oi |i ∈ I))
=∩(Oi |i ∈ I ) in Y. Thus Y is α-I- Alexandorff. Thus Y is α
Alexandorff.
Theorem 2.20 If X and Y are α-I-Alexandorff, then X ×Y
is α-I-Alexandorff.
Proof. Let {Wi : i ∈ I } be a collection of open sets in X ×Y
and let W be its intersection. Suppose that W is not α-Iopen. Then there exists (x, y) ∈ W and open sets U ⊆ X , V
⊆ Y with (x, y) ∈ U ×V and (U ×V )∩intW = ∅. For
each i ∈ I there exist open sets Oi ⊆ X , Ri ⊆ Y such that
(x, y) ∈ Oi × Ri ⊆ Wi . Since X and Y are α-I-Alexandorff,
then there exist non-empty α-I-open sets G ⊆ X and H ⊆ Y
with G ⊆ U ∩(∩i∈I Oi ) and H ⊆ V ∩(∩i∈I Ri ). Clearly G × H ⊆
(U × V ) ∩ intW , a contradiction. Thus X × Y is α-IAlexandorff.
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