Rev. Acad. Canar. Cienc., XVIII (Núms. 1-2), 23-31 (2006) (publicado en agosto de 2007)
y-CONVERGENCE OF NETS AND FILTERS
S. Ganguly & Ritu Sen*
Department of Pure Mathematics, University of Calcutta
35, Ballygunge Circular Road, Kolkata-700019
India
e-mail: gangulys04@yahoo.co.in
Abstract : In this paper we discuss the notion of ¡ - convergence of nets and filters. We
also introduce the concept of ¡- compact spaces and characterize such spaces.
AMS Subject Classification : 54C35
Keywords : ¡ - open sets, ¡ - continuous functions, ¡ - irresolute functions, ¡ - convergence
of nets and filters, ¡ - compact spaces.
1 Introd uction
In [5] Min first introduced the notion of ¡- open sets ( originally called ¡- sets). In the
same paper Min has also introduced and discussed the notion of ¡ - continuous functions
and ¡- irresolute functions. Later in [1] Ganguly and Sen have introduced the concept of ¡convergence
of nets.
The aim of this paper is to introduce ¡ - convergence of filters and discuss ¡ - convergence
of nets and filters.
Throughout this paper, (X, T) (simply X) and (Y, T*) always mean topological spaces.
Let S be a subset of X. The closure (resp. interior) of S will be denoted by cl(S) (resp.
int(S)).
A subset S of X is called a semi-open set [4] if S ~ cl(int(S)). The complement of a semiopen
set is called a semi-closed set. The family of all semi-open sets in a topological space
*The second author is thankful to CSIR, India for financia! assistance.
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(X, T) will be denoted by SO(X). A subset M(x) of a space X is called a semi-neighborhood
of a point x E X if there exists a semi-open set S such that x E S ~ M(x). In [2] Latif
introduced the notion of semi-convergence of filters. Let S(x) ={A E SO(X) : x E A} and
let Sx = {A ~ X : there exists μ ~ S(x) such that μis finite and nμ ~ A}. Then Sx is
called the semi-neighborhood filter at X . For any filter r on X we say that r semi-converges
to X if and only if f is finer than the semi-neighborhood filter at X.
Definition 1.1 [3] A subset U of X is called a ¡ open set if whenever a filter r semiconverges
to x and x E U, U E r. The complement of a¡- open set is called a ¡- closed
set.
The intersection of all ¡ - closed sets containing A is called the ¡ - closure of A, denoted
by cl,,(A). A subset A is¡- closed iff A = cl1 (A). We denote the family of all ¡ - open sets
of (X,T) by T1. It is shown in [5] that T1 is a topology on X. In a topological space (X,T) ,
it is always true that T ~ S(X) ~ T 1 .
Definition 1.2 [5] A function f : X ---+ Y is said to be ¡ - continuous if the inverse image
of every open set of Y is ¡- open in X .
The set of ali ¡ - continuous functions from X into Y is denoted by ¡C(X, Y).
Definition 1.3 [5] A function f : X ---+ Y is said to be ¡ - irresolute if the inverse image of
every ¡ - open set of Y is ¡ - open in X.
Definition 1.4 A net {x>. : >. E A} in X is said to converge to a limit x E X (in symbol,
X>. ---+ x) if for every neighborhood V of x, :J a >.0 E A such that >. 2: >.0 implies X>. E V.
Definition 1.5 [1] A net {x>. : >. E A} in X is said to ¡ - converge to a limit x E X (in
symbol, x >. ---+ 1 x) if for every ¡ - open set V containing x, :J a >.0 E A such that >. 2: >.0
implics X>. E V.
Definition 1.6 Let r be a filter on a topological space X . Then r is said to converge to X
if each neighborhood of X is a member of r, i.e., Nx ~ r .
2 ry- convergence of nets and filters
In this article we first discuss the ¡- convergence of nets. Then we discuss the ¡- convergence
of filters.
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Definition 2.1 Let X be a topological space. A subset A~ X is said to be a 1- neighborhood
of a point x E X if and only if there exists a 1- open set U such that x E U ~ A.
The set of all 1- neighborhoods of a point x E X is denoted by 1Nx.
Theorem 2.2 [1] A function f : X -t Y (where X and Y are topologicaJ spaces) is 1-
irresoJute at a point x E X iff for any net {x.A : >- E A} in X 1- converging to x, the net
{f(x.A): >-E A} 1- converges to f(x) in Y.
Theorem 2.3 A function f : X -t Y is 1- continuous at a point x E X iff for any net
{x.A: >-E A} in X 1- converging to x, the net {f(x.A) : >-E A} converges to f(x) in Y.
Proof : First assume that f is 1- continuous at x E X. Let { x .A : >- E A} be a net in X 1-
converging to x. Let V be an open set in Y containing f(x). Now there exists a 1- open set
U containing x in X such that f(U) e V. Now {x.A: >-E A} 1- converges to x impJies that
there exists a .\0 E A such that X>. E U for all A ~ .\0 . Hence, for all >- ~ .\0 , f(x>.) E V.
This shows that {f(x>.): >-E A} Jies eventua!Jy in V and hence it converges to f(x).
ConverseJy, Jet f be not 1- continuous at x. Then there exists an open set W containing
f(x) in Y such that from every 1- open set U containing x E X, there exists an eJement xu
with f (xu) (j_ W. Let 1Nx be the 1- neighborhood system at x. So, { xu : U E 1Nx} is a
net in X 1- converging to x, but the net {f(xu) : U E 1Nx} in Y does not lie eventually in
W and consequentJy it cannot converge to f(x) .
Next we form the 1- neighborhood filter at a point x E X and discuss the notion of 1-
convergence of filters.
Definition 2.4 Let (X,T) be a topoJogicaJ space. For x E X , Jet 1(x) ={A~ T"I: x E A}.
Then 1(x) has the finite intersection property. Thus 1(x) is a filter subbasis on X . Let lx
be the filter generáted by 1(x), i.e., lx ={A~ X : there exists μ ~ T"I such that μis finite
and nμ ~A}. lx will be called the 1- neighborhood filter at x.
Definition 2.5 Let r be a filter on (X, T). r is said to 1- converge to X E X iff r is finer
than the 1- neighborhood filter at x.
Definition 2.6 Let r be a filter on (X, T). r is said to 1- cluster to X E X iff every FE r
intersects each A E 1(x).
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Example 2.7 Consider X= {1, 2,3, 4}, T = {<I>,X, {l}, {l, 2} , {1, 2, 3}}. Then T'Y =
{<I>,X, {l}, {l, 2}, {l, 3}, {l, 4}, {1, 2, 3}, {l,2,4}}. Then the ¡ - neighborhood system at 4
consists of {l, 4}, {l, 2, 4} and X. Now, ¡ (4) = { {l, 4} , {l, 2, 4}, X} and ¡ 4 = { {l, 4} , {l, 2, 4}, X}.
Then ¡ 4 is the ¡ - neighborhood filter at 4.
Consider the filter r = {{l, 4},{1, 2,4}, {l, 3,4},X}. Then r is finer than ¡ 4 and hence r
¡ - converges to 4. Also every FE r meets each A E ¡(4). Hence r ¡ - clusters at 4.
Proposition 2.8 Let r be a filter on (X, T) ¡ - converging to X in X. Then r also converges
to X.
Proof : Obvious.
Proposition 2.9 Let r be a filter on (X , T) which ¡ - clusters to X in X. Then X E n{ cl-y(F) :
FE r}.
Proof : Obvious.
Definition 2.10 Let (X, T) be a topological space. Let r be a filter on X. Then a point
X E X is said to be a strong ¡- cluster point of r iff every FE r intersects each A E lx·
Proposition 2.11 If risa filter on (X, T) strongly ¡ - clustering to X E X , then r also ¡ clusters
at x.
Proof : Obvious.
That the converse may not be true is proved by the following example.
Example 2.12 Consider X= {1,2,3,4}, T = {<I>,X, {2,3}, {4}, {2, 3,4}}. Then the ¡ open
sets are <I>, X , {2, 3}, { 4}, {2, 3, 4}, {l, 4} , {l, 2, 3}. Then ¡(1) = { {l, 4}, {l, 2, 3}, X}
and the ¡ - neighborhood filter at 1 is ¡ 1 = {{l, 4} , {1,2, 3} , {l},X}. Consider the filter
r = {{2,4}, {l ,2, 4} , {2,3, 4}, X}. Clearly r ¡- clusters to 1 since every FE r meets
each A E ¡ (1). But f <loes not strongly ¡ -cluster to 1 since {2,4} E r , {l} E ¡ 1, but
{2,4} n {l} = <f>.
Definition 2.13 Let f : X -+ Y be a function and r be a filter on X. Let f(f) = {f(F) :
F E r}. Then f(f) may not be a filter on Y , but it is a base for sorne filter on Y and this
filter is denoted by the symbol P(r) and is called the image of the filter r under the map f.
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Theorem 2.14 Let f: X-+ Y be a function where X and Y are topological spaces. Then
f is ¡ - continuous at x E X iff for any filter r on X ¡ - converging to x, the filter ¡P(r) on
Y converges to f(x).
Proof : First assurne that f is ¡ - continuous at x. Let r be a filter on X ¡- converging to
x. Let W be a neighborhood of f(x) in Y. Then there exists a¡- open set V containing x
in X with f(V) e W. Now since r ¡ - converges to x, so V E r and hence f(V) E f(f) =}
W E ¡tt(r). So, J"(r) converges to f(x).
To prove the converse, assurne that the given condition holds. Let W be a neighborhood
of f(x) in y and r be the filter of all ¡ - neighborhoods of X in X. Then clearly r ¡ - converges
to x and hence by the assurned condition, the filter J"(r) on Y converges to f(x) and so
W E ¡tt(r). Now W E JP(r) =} W::) f(V) for sorne¡- neighborhood V of x, i.e., W::) f(V)
for sorne¡- open set V containing x. This shows that f is ¡ - continuous at x.
Theorem 2.15 Let f : X-+ Y be a function where X and Y are topological spaces. Then
f is ¡ - irresolute at x E X iff for any filter r on X ¡ - converging to x, the filter J"(r) on Y
¡ - converges to f ( x).
Proof : Let f be ¡- irresolute at X and let r be a filter on X ¡ - converging to X. Let V
be a¡- open set containing f(x). Since f is ¡ - irresolute at x, there exists a¡- open set
u containing X such that f(U) e v. Now since r ¡ - converges to x, u E r and hence
f(U) E f(f) =}V(:) f(U)) E JP(r). Thus the filter ¡tt(r) on Y ¡ - converges to f(x).
Conversely, let the given condition holds. Let V be a¡- open set in Y containing f(x)
and r be the filter of all ¡ - neighborhoods of x in X. Then clearly r ¡ -converges to x and
by the given condition, the filter JP(f) on Y ¡ - converges to f(x) and hence V E J "(r). This
irnplies that V::) f (U) for sorne¡- open set U containing x. Hence the function f: X-+ Y
is ¡ - irresolute at x .
3 Characterizations of 1- compact spaces
In this article we first introduce the notion of ¡ - cornpact spaces and then try to characterize
such spaces.
Definition 3.1 A topological space (X, T) is called ¡- cornpact if every ¡-open cover of X ,
i.e., a cover of X by ¡- open sets in X has a finite subcover.
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Result 3.2 It has been shown in [3] that ¡ - compactness implies compactness and Latif
has given an example to show that the converse may not be true.
Definition 3.3 Let {xA : A E A} be a net in X. Then the family of tails {TA : >. E A}
where, TA = {Xμ : μ :;::: A} is a base for sorne fil ter on X. This fil ter is called the fil ter
generated by the net {xA: >.E A}.
Definit ion 3.4 Let r be a filter on X. Let Dr = {(x, F) : x E F E r}. If we set for any
two (x, F) , (y, G) E Dr, (x, F) :;::: (y, G) iff F ~ G, then (Dr, :;:::) becomes a directed set.
Then the map Pr : Dr ---+ X : (x, F) ---+ x is a net in X which is called the net defined by
the filter r.
Theorem 3.5 Let (X, r) be a topological space. A net {xA : >. E A} in X ¡ - converges to
x in X iff the filter generated by the net ¡ - converges to x.
Proof : Let the net {xA: >.E A} in X ¡ - converges to x. We have to show that the filter
r generated by the net also ¡- converges to x. Choose any ¡ - neighborhood U of x. Since
{xA : >.E A} ¡ - converges to x, there exists a .\0 E A such that xA E U, for all >.:;::: .\0 , i.e.,
TAo E U. Since TAo is the base for the filter r, so TAo E r. Hence u E r. Thus the filter r
generated by the net {xA: >.E A} ¡ - converges to x.
Conversely, let the filter r generated by the net {xA : A E A} ¡ - converges to X. Let u
be a¡- neighborhood of X. Since r ¡ -converges to x, u E r. Now since {TA : A E A} is a
base for r, there exists a Ao E A such that TAo ~u, i.e., {xA : A::::: >-o} ~u. Thus XA E u,
for all >. :;::: .\0 . Hence the net { x A : A E A} ¡ - converges to x.
Theorem 3.6 Let (X, r) be a topological space. A filter ron X ¡ - converges to x iff the
net Pr : Dr ---+ X defined by the filter ¡ - converges to x.
Proof : Let r ¡- converges to X. Choose a ¡ - neighborhood u of X. Then u E r and
so (x, U) E Dr. Let (y, V) E Dr be such that (y, V) :;::: (x, U). Then y E V ~ U, i.e.,
Pr(y, V) E U. This shows that the net Pr líes eventually in U and hence the net Pr ¡ converges
to x .
Conversely, let the net Pr : Dr ---+ X generated by the filter r ¡-converges to X . Let u be
a ¡- neighborhood of x. Then the net Pr lies eventually in U, i.e., there exists (y, V) E Dr
such that for all (z, G) :;::: (y, V) , Pr(z, G) = z E U. In particular, for any point k E V,
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(k, V) :'.". (y, V), so that k E U. This shows that V~ U. But (y, V) E Dr =;. V E r =;. U E r.
Hence the filter r ¡- converges to X.
Definition 3.7 A net {x.x: A E A} in (X,1) is said to¡- cluster ata point x E X if given
a¡- neighborhood U of x and μE A, there exists a>- E A such that x.x E U, for ali >- :'.". μ.
We say that the net líes frequently in U.
Theorem 3.8 Let (X, 1) be a topological space. A net { x.x : >- E A} in X ¡ - clusters at a
point x E X iff the filter generated by the net ¡- clusters at x .
Proof: Suppose {x.x : A E A}¡- clusters at X. Let u be a¡- neighborhood of x. Let r be
the filter generated by the net {x.x : >-E A} and FE r . We have to show that Un F # </>.
If possible, !et Un F = </>. Since F E r , there exists a .>-0 E A such that T.x0 ~ F, i.e.,
{x.x : >- :'.". .>-0} ~ F. Therefore, {x.x : >- :'.". .>-0} n U=</>. Hence for .>-0 E A, there <loes not
exist A E A such that x.x E U, for A:'.". .>-0 . This implies that {x.x: >-E A} <loes not ¡-cluster
at x, a contradiction. Hence un F # </> and thus the filter r ¡ - clusters at X.
Conversely, let the fil ter r generated by the net { X>. : A E A} ¡ - clusters at X. Let u be
a¡- neighborhood of X and >-o E A. Since r ¡- clusters at x, un F # </>, for ali FE r. Now
T.x0 E r =;. T.x0 n U#</>=;. there exists >-E A such that x.x E U for >-:'.".>-o=;. {x.x : >-E A}
¡- clusters at x.
Theorem 3.9 Let (X, 1) be a topological space. A filter ron X¡- clusters at X iff the net
Pr : Dr -> X defined by the filter ¡ - clusters to x.
Proof : Let r ¡- clusters at X. Let u be a ¡ - neighborhood of X and (y, F) E Dr. Then
y E FE r. Since r ¡ - clusters at x, F n U#</> and so we can choose a point z E F n U.
Then (z, F) E Dr and (z, F) :'.". (y, F) implies that Pr(z, F) = z E U. This shows that the
net Pr is frequently in U and hence Pr ¡- clusters at x.
Conversely, !et the net Pr : Dr -> X ¡- clusters at x. Given any ¡- neighborhood U of
x, !et G be an arbitrary member of r. We have to show that Un G # <f>. Choose any y E G
[since GE r, G # </>]. Since Pr ¡- clusters at x and (y,G) E Dr, there exists (z,H) E Dr
with (z, H) :'.". (y, G). Therefore, z EH~ G such that Pr(z, H) = z E U. Hence G n U#</>.
Thus the filter r ¡ - clusters at X.
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Theorem 3.10 A topological space (X, T) is ¡ - compact iff for every collection of ¡ - closed
sets {F"': a E A} in X possessing finite intersection property (in short, f.i.p. ), the intersection
íl F"' of the entire collection is nonempty.
0tEA
Proof: Let (X,T) be ¡- compact and let {F"' : a E A} be a collection of ¡ - closed sets
having f.i.p. If possible, let íl F"' = cp. Then X = X \ íl F"' = LJ (X \ F"'). Thus
0tEA aEA 0tEA
{X \ Fa : a E A} is a ¡- open cover of X. Thus there exists a finite subset A0 of A su ch that
X = LJ (X\ F"') =X\ íl Fa, i.e., íl F"' =e/>, a contradiction. Hence íl F"' =f cp.
0tEAo 0tEAo aEAo 0tEA
Conversely, let {Ga: a E I} be a ¡ - open cover of X. Then cp =X\ LJ Ga = íl (X\ Ga)·
aEI 0tEI
Since {X \ G"' : a E I} is a collection of ¡ - closed sets, by hypothesis, it cannot have f.i.p.
Thus there exists a finite subset !0 of I such that íl (X\ G"') = cp, then X = LJ Ga,
0tElo 0tElo
proving that X is ¡ - compact.
Theorem 3.11 A topological space (X, T) is ¡ - compact iff every filter in X has a¡- cluster
point in X.
Proof : First let X be ¡- compact. Let r be a filter on X. Then cL.¡f = { cl"YF : F E r} is
a family of ¡ - closed subsets of X with f.i.p. Since X is¡- compact, íl cl"YF =f cp. Take a
FEr
point X E n cl"YF. Then X is a¡- cluster point of r.
FEr
To prove the converse, !et the condition holds. Let { G"' : a E I} = B be a family
of ¡ - closed sets in X with f.i.p. Now there exists a filter r on X such that B ~ r. By
the assumed condition, r ¡- clusters at X E X. Then X E cl"YF, for all F E r. But each
G"' E f::? x E cl'Y(G"') , for all a E l , i.e., x E Ga, foral! a E l . So íl G"' =f cp. Hence X is
0tEI
¡ - compact.
With the help of the above proved theorems, we can now state that :
Theorem 3.12 The following conditions are equivalent for a topological space X :
(a) X is¡- compact.
(b) Every filter in X has a ¡ - cluster point.
( c) Every net in X has a ¡ - el uster point.
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References
[1] S. Ganguly and Ritu Sen ; The ¡ - open open topology for function spaces, communicated.
[2] R. M. Latif ; Semi convergence of Filters and Nets, Math. J. of Okayama University, 4
(1999), 103- 109.
[3] R.M. Latif ; Charncterizations and Applications of ¡ -open sets, accepted for publication
in Arab Journal of Mathematical Sciences.
[4] N. Levine ; Semi open sets and semi continuity in Topological spaces, Amer. Math.
Monthly, 70 (1963), 36- 41.
[5] W. K. Min ; ¡- sets and ¡- continuous functions, Int. J. Math. Math. Sci. , 31 (2002),
no 3, 177- 181.
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