Vol. 5, No. 1, 2017, 11-15 ISSN: 2321 – 9238 (online) Published on 16 March 2017 www.researchmathsci.org
DOI: http://dx.doi.org/10.22457/pindac.v5n1a2
11
Progress in
Weak Convergence of Filters
A.A.Hakawati1, B.Manasrah2 and M.Abu-Eideh3
1
Department of Mathematics, An-Najah National University Nablus, Palestine. E-mail: [email protected]
2
Department of Mathematics, Hebron University, Palestine. E-mail: [email protected]
3
Department of Mathematics, An-Najah National University Nablus, Palestine. E-mail: [email protected]
Received 24 February 2017; accepted 16 March 2017
Abstract. In this paper, we introduce what we called weak convergence of filters and show that, in Uryson spaces, weak limits are unique. Moreover, we show that, in a
regular space X, with E⊆X , x∈Eif andonlyif there is a filter ℑon X which converges weakly to xand F∩E ≠
φ
∀F∈ℑ .We also prove that closure continuous maps preserve weak convergence of filters. As a main result, we prove that, in regular spaces, weak convergence of filters is equivalent to convergence of filters.Keywords: Weakly convergent filter 1. Introduction
The concept of weak convergence of sequences and nets was studied long time ago [1,2]. In this paper, we study weak convergence of filters and obtain some useful results. In particular, and among other results, we prove here, that in regular spaces, convergence and weak convergence of filters are equivalent.
A filter ℑ/ is said to be finer than the filter ℑif , / / / .
F F that such F
F all
for ∈ℑ∃ ∈ℑ ⊆ This is written ℑ⊆ℑ/ ,ℑ/ ≥ℑ. or
Let Ux be the neighborhood system of x in a topological space X on which a filter ℑis
given. Of course, Ux is a filter on X. We say that the filter ℑconverges to x,
. write
we
and ℑ→x if ℑ≥Ux
If
β
is a filter base on X, then the family ℑ={
F :F⊇ BforsomeB∈β
}
is the filtergenerated by
β
. If ℑis a filter on a topological space X then, by ℑ we mean the filtergenerated by the filter base
β
={
F :F∈ℑ}
, where Fstands for the closure of F. It isA.A. Hakawati, B. Manasrahand M. Abu-Eideh
12
A function f :X →Y is said to be closure continuous at x°∈X if for every neighborhood Vof f(x)thereisaneighborhoodU of x° with f(U)⊆V. If this condition is satisfied at each point of X then f is called closure continuous on X [1]. It is not hard to show that continuous functions are closure continuous, for if V is a neighborhood of f(x) take a neighborhood U of x such f(U)⊆V, and so f(U)⊆V . Now, by continuity of f, f(U)⊆ f(U)⊆V.
The converse, however is untrue.
For example, take X =
{
a,b,c}
,τ
1={
φ
,X,{ }{ }
a , a,b}
,τ
2={
φ
,X,{ }{ }
b, b,c}
.Then take the identity function i:
(
X,τ
1) (
→ X,τ
2)
which is discontinuous, because{ } { }
b c bc i−1( , )= ,which is not open in
τ
1. In the meantime i is closure continuous. Tosee this, for any x∈X and any neighborhood V of f(x) , V =X
2
τ , and so, for any
neighborhood U of x we have:
( )
2
τ V X U
f ⊆ = .
Finally, a space X is called a Uryson space if whenever x1≠x2in X,there are open sets U and V in X containing x1 and x2 respectively, suchthat U∩V =
φ
[3].We close this introduction by noting that topologists nowadays prefer to insert their applications in generalized, or enlarged settings. This might help escaping tight limits and specifications. For this we refer interested readers to compare with [6,7]. In specific One can consult [8] for the general setting of Rough Set Theory.
2. Weak convergence of filtres
Definition 2.1. If ℑis a filter on a topological space X and x∈X , then ℑ is said to
converge weakly to x ( written ℑ→w x ) if ℑis finerthanUx.That is,
{
x}
xx whereU isthe filter generated by thecollection U U U
U ∈
≥
ℑ : .
Remark 2.2. It is easy check to see that Ifℑ→xthen ℑ→w x but not conversely. The following example shows this.
Example 2.2. Let X =
{
a,b,c}
with the topologyτ
={
φ
,X,{ } { } { }
a, b, a,b}
. Consider the filter ℑ={
X,{ }
a,c}
.Now, ℑ→w a since the neighborhood system Ua =
{
{ }{ }{ }
a, a,b, a,c ,X}
, andUa =
{
{ }
a,c,X}
, from which it clear that ℑ≥Ua .13
Theorem 2.3. Let f :X →Y be a function, and let x∈X . Then f is closure continuous at xif and only if, whenever ℑis any filter in X with ℑ→w x in X , we have : f(ℑ)→w f(x) in Y.
Proof: Suppose f is closure continuous at x, and suppose ℑ→w x. Let Uf( x)be the neighborhood system of f(x).
We claim that f(ℑ)≥Uf(x) .
For this, let V∈Uf( x) be arbitrary.
Now, Visaneighborhoodof f(x), and since f is closure continuous at x, there is a neighborhood Uof x suchthat f(U)⊆V. Thus, f(U)∈ f(ℑ). Hence, f(ℑ)≥Uf(x),
which means that f(ℑ)→w f(x).
Conversely, suppose that whenever ℑ→w x,wehave: f(x)→w f(x).
Let Vbeaneighborhoodof f(x) . Since Ux≥Ux,Ux→w x. By hypothesis, we
have: f(Ux)→w f(x) . Thus f(U)≥Uf(x) . Therefore, there is a neighborhood
V U f that such x of
U ( )⊆ , and hence, f is closure continuous at x.
Theorem 2.4. Let Xbe a Uryson's space, and ℑbe a filter on X. If ℑ→w xand ℑ→w y,then x=y.
Proof: Suppose that xand yare distinct elements in X. Since Xis a Uryson's space,
choose neighborhoods U andV of xand yrespectivelywithU∩V =
φ
.Now, since ℑ→w x we have: ℑ≥Ux and U∈ℑ.
Also, since ℑ→w y we have: ℑ≥Uy and V∈ℑ.
Thus,
φ
=U∩V∈ℑwhich contradicts the fact that ℑis a filter on X .Therefore x and ycould not have been distinct. We close this section with the following Remark.
Remark 2.5. If ℑand ℑ/ are filters on Xwith ℑ/ ≥ℑand if x
then
x w
w ℑ →
→
ℑ /
, as well.
Proof: Since ℑ→w x, then ℑ≥Ux. But since ℑ/ ≥ℑwe have that ℑ/≥Ux .
Thus, ℑ/→w x.
3. Main results
A.A. Hakawati, B. Manasrahand M. Abu-Eideh
14
Definition 3.1. Let X be a topological space , and let x∈X and E⊆ X. Thenx is
called a weak-closure point of Eif for all neighborhoods Uof x,U∩E≠
φ
.The set of all weak closure-points of Eis denoted, here, by Ew. It needs only a quick observation to see that the following Lemma is a true statement.
Lemma 3.2. Let Ebe a subset of a topological space X . Then E⊆Ew. The converse of this lemma is false and here is an example.
Let X =
{
a,b,c,d}
with topologyτ
={
φ
,X,{ }{ }{ }{ }{
a , a,b, c, a,c, a,b,c}
}
.Let E=
{ }
b . Then E={ }
b,d ,but Ew ={
a,b,d}
.Theorem 3.3. Let Ebe a subset of a topological space X and let x∈X. Then X
on filter a is there if only and if E
x∈ w ℑ such that ℑ
∈ ≠
→
ℑ w xand F∩E
φ
for all F .Proof: Suppose x∈Ew. So, for all neighborhoods U of x,U∩E≠
φ
. Consider ℑ=Ux. It is clear that U xw
x→ . Moreover, we have: for all U∈Ux ,U∩E≠
φ
.Conversely, let ℑbe a filter on Xwith ℑ→w xand F∩E≠
φ
for all F∈ℑ.We must show that x∈Ew.
Let Ube a neighborhood of x. Then ℑ≥Ux becauseℑ→w x.
But then by hypothesis,U∩E≠
φ
. Therefore x∈Ew.For our next result, we need to recall the following definition from [4].
Definition 3.4. A space Xis said to be regular if ℑ→x whenever ℑ→x.
Lemma 3.5. Let Xbe a regular space, and let ℑbe a filter on X and x∈X. If ℑ→w x then ℑ→x.
Proof: Suppose ℑ→w x . Then, ℑ≥Ux.
But Ux →xand X regular, so Ux→x.Therefore, ℑ→x. Combining Remark (2.5) and lemma (3.5), One gets the following :
Theorem 3.6. Let Xbe a regular space, and ℑbe a filter on X. Then, ℑ→xif and onlyif ℑ→w x.
We close this paper with the following theorem.
15 Proof: By lemma (3.2)E⊆Ew.
For the other inclusion, let x∈ Ewbe arbitrary. Then by theorem (3.3), X
on filter a is
there ℑ such that ℑ→w xand F∩E≠
φ
for all F∈ℑ. But since Xis regular, by lemma (3.5) ℑ→xand F∩E≠φ
for all F∈ℑ.Hence, x∈E[3, Theorem 12.6].
Thus, Ew ⊆E. Thus equality is achieved.
Acknowledgment. The authors would like to thank Muath Karaki for his final touches and arrangements of the article.
REFERENCES
1. D.R.Andrew and E.K.Whittlesy, Closure continuity, American Monthly, 73 (1966) 758-759.
2. A.A.Hakawati and B.A.Manasrah, Weak convergence of nets, Islamic University Journal, 5(1) (1997) 45-55.
3. S.Wilard, General Topology, Addison-Wesely Publishing Company, 1970. 4. H.J.Kowalsky, Topological Spaces, Academic Press, N.Y. and London, 1965.
5. T.Indira and S.Geetha, Alpha closed sets in topological spaces, Annals of Pure and Applied Mathematics, 4(2) (2013) 138-144.
6. J.Thomas and S.J.John, Properties of Dµ-compact spaces in generalized topological spaces, Annals of Pure and Applied Mathematics, 9(1) (2015) 73-80.
7. B.P.Mathew and S.J.John, Some special properties of I-rough topological spaces, Annals of Pure and Applied Mathematics, 12 (2) (2016) 111-122.
