Download ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM

DOI: 10.2478/v10127-012-0023-y
Tatra Mt. Math. Publ. 52 (2012), 29–45
ON THE I-R0 SPACES
AND I-STRONG QUASI-UNIFORM CONVERGENCE
Robert Drozdowski — Agata Sochaczewska
ABSTRACT. Using ideals, we introduce a new class of topological spaces that
are more general concept than R0 spaces [Császár, Á.: General Topology. Akadémiai Kiadó, Budapest, 1978]. Different properties will be studied for that class
of topological spaces. We also define a new kind of convergence for sequences of
functions and formulate more general results to those contained in [Sochaczewska,
A.: The strong quasi-uniform convergence, Math. Montisnigri 15 (2002), 45–55].
1. Introduction
Let (X, τ ) be a topological space. For a set, A ⊂ X the symbols clτ (A) and
intτ (A) stand for the closure and the interior of the set A, respectively. If A ⊂ X
is open (dense, nowhere dense etc.) with respect to the topology τ, we will say
that A is τ -open (τ -dense, τ -nowhere dense, etc., respectively).
A topological space (X, τ ) will be called T3 space if, for each x ∈ X and for
each closed set F such that x ∈
/ F , there exist open sets U, V such that x ∈ U ,
F ⊂ V and U ∩ V = ∅ [2].
In this article, the undefined terms and symbols should be understood in the
sense of [3].
According to [2], [7], a topological space (X, τ ) is said to be
R0space if for each
x ∈ X and each τ -open set G that contains x, we have clτ {x} ⊂ G. R0 spaces
were used to characterize axioms of separations for topological spaces, i.e., they
were introduced to characterize T1 spaces. Now, it is well-known that a topological space (X, τ ) is T1 space if and only if it is T0 space and R0 space [2], [7].
An example of R0 space that is not T1 space can be found in [2]. Remark that
c 2012 Mathematical Institute, Slovak Academy of Sciences.
2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary: 54A20; Secondary: 54B10,
54B05.
K e y w o r d s: quasi-strong convergence, I-open covers, continuous function, I-T3 space,
I-R0 space.
29
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
notations of R0 spaces and T0 spaces are independent, i.e., there exists a topological space that is T0 but not R0 space, and conversely. For example, the space
of reals with the left topology τl is T0 space that is not R0 space whereas the
space of reals with the symmetric topology τs is R0 space that is not T0 space.
However, in the literature one can find a lot of topological spaces (X, τ ) that
do not satisfy the condition formulated for R0 spaces. It is sufficient to take
property: for each x ∈ R
into account the space (R, τl ) which has the following
and each τl -neighborhood G of x, the set clτl {x} \ G is τl -nowhere dense. This
observation leads us to the definition of I-R0 spaces.
Let us recall, a family I of subsets of a nonempty set X is said to be an ideal
if it satisfies the following conditions:
(a) if A ∈ I and B ⊂ A, then B ∈ I,
(b) if A, B ∈ I, then A ∪ B ∈ I [2].
1.1 Let I be an ideal of subsets of a set X. A topological space
(X, τ ) is said to be I-R0 space if, for each x ∈ X and each open set G such that
x ∈ G, the set clτ {x} \ G belongs to I.
Examples given below show that an ideal I of subsets of X and a topology τ
in X play an important role in the above definition.
Example 1.1. Let X = {a, b, c}, and consider topologies
and τ2 = ∅, X, {c}, {b, c} .
τ1 = ∅, X, {a}, {a, b}
Let I = ∅, {b}, {c}, {b, c} . One can
that (X, τ1 ) is I-R0 space whereas
show
/ I.
(X, τ2 ) is not I-R0 space, since clτ2 {b} \ {b, c} = {a} ∈
Example 1.2. Let X = R be ordered by relation “” defined in the following
way
x y ⇐⇒ ∃k∈Z (y = kx).
Consider
the topology τL introduced in X by the system of neighborhoods
{L(x)} x∈X , where L(x) = {y : y x}, for each x ∈ X (for details, see [3]).
Let I1 be an ideal of countable sets and I2 an ideal of finite sets in X. If x ∈ X,
then
clτL {x} = {y ∈ X : x y} = {kx : k ∈ Z} .
Now, if G
is a neighborhood of x ∈ X, then clτL {x} \ G is at
most countable,
.
On
the
other
hand,
cl
{x}
\
G
belongs
to
I
{1}
= Z and L(1) =
i.e.,
cl
1
τL
1 τL
k : k ∈ Z \ {0} . By this,
1
clτL {1} \ L(1) = Z \
: k ∈ Z \ {0} = Z \ {1, −1} ∈
/ I2 .
k
It has shown that (X, τL ) is I1 -R0 space that is not I2 -R0 space.
30
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
Now, for a point x ∈ X and a cover α of a topological space (X, τ ), St(x, α)
denotes
the star of x with respect to the cover α, i.e., a set of the form St(x, α) =
{U ∈ α : x ∈ U } [3].
Let (fn )∞
n=1 be a net of functions and f be a function, where fn , f are defined
on a topological space (X, τX ) with values in a topological space (Y, τY ).
1.2 ([9]) A net (fn )∞
n=1 is said to be strongly quasi-uniformly
convergent to f if for each open cover α of (Y, τY ) and for each x0 ∈ X there
n≥
exists n0 ∈ N such that for each
n0 there exists a neighborhood U of x0
with the property fn (x) ∈ St f (x), α for all x ∈ U .
Strongly quasi-uniform convergence of a net of functions was given in [9]. It is
a generalization of the notation of a strong convergence defined by I. K u p k a
in [6].
1.3 ([6]) A net (fn)∞n=1 is said to be strongly convergent to a func-
n0 ∈ N such that for each
tion f if for each open cover α of (Y, τY ) there exists
n ≥ n0 and for every x ∈ X we have fn (x) ∈ St f (x), α .
I. K u p k a proved that if (fn )∞
n=1 is a net of functions defined on a topological space with values in a T3 space (Y, τY ) for each n ∈ N and (fn )∞
n=1 is strongly
convergent, then it is pointwise convergent. In [9], it was pointed out that we can
substitute the strong convergence of a net with a strong quasi-uniformly convergent net of functions with values in a T3 space. The last result was extended
in [10], namely
1.1 ([9])
Let fn , f : X → Y and let Y be an R0 space. If the seis strongly quasi-uniformly convergent to a function f, then it is
quence
pointwise convergent.
(fn )∞
n=1
By the above theorem, a strongly quasi-uniform convergence net of functions
with values in R0 space is stronger than a pointwise one but weaker than a uniform convergence. I. K u p k a [6] proved that the limit of strong convergent
sequence of continuous functions with values in a T3 space is continuous as well.
This shows that some weaker kinds of convergence of nets of continuous functions imply continuity of their limits as well as uniform convergence of nets of
continuous functions. The same property also preserves a quasi-uniform convergent net of functions in the sense of A r z e l a [1] or a net of functions convergent
in the sense of P r e d o i [8]. A similar result was given in [9].
1.2 ([9]) Let fn, f : X → ∞Y, fn be continuous for each n ∈ N and let
Y be a T3 space. If the sequence (fn )n=1 is strongly quasi-uniformly convergent
to f, then f is continuous as well.
Here, we introduce notation of I-strongly quasi-uniformly convergence and
formulate more general theorems than Theorem 1.1 and Theorem 1.2. Although
31
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
this kind of convergence will be stronger than the strongly quasi-uniform convergence, our results will be formulated for a wider class of topological spaces Y.
2. I-R0 spaces
In this section, we investigate the basic properties for I-R0 spaces (see Definition 1.1), i.e., we formulate the sufficient and necessary conditions for I-R0
spaces; further we describe some operations for such a kind of topological spaces.
2.1 Let I1 and I2 be ideals of subsets of a nonempty set X such
that I1 ⊂ I2 . If (X, τ ) is an I1 -R0 space, then it is I2 -R0 space.
P r o o f. If x ∈ X and G is a τ -open set which contains x, then clτ ({x}) \ G ∈ I1 .
Thus, clτ ({x}) \ G ∈ I2 , i.e., (X, τ ) is I2 -R0 space.
2.2 Let (X, τ ) be a topological space, I be an ideal of subsets of X
and N be an ideal of nowhere dense sets with respect to topology τ. If (X, τ ) is
I-R0 space, then it is N-R0 space as well.
P r o o f. Let us assume that (X, τ ) is I-R0 space. It is sufficient to see that,
forevery
x ∈ X and every open set G such that x ∈ G, the set of the form
clτ {x} \ G is τ -nowhere dense.
2.1 If (X, τ ) is an R0 space, then it is I-R0 space for an arbitrary
ideal I of subsets of X.
Obviously, there exists a topological space that is not R0 space but I-R0 space
with respect to an ideal I of subsets of X. The following example shows it.
Example 2.1. Let us consider the left topology τl on the real line and let I be
an ideal of τl -nowhere dense sets. If G is a τl -open set and x ∈ G then one can
find a set of the form V = (−∞, b) such that x ∈ V, V ⊂ G, and b ∈ R. By this,
clτ {x} \ G ⊂ clτ {x} \ V = [b, ∞).
Evidently, [b, ∞) belongs to I, so the space of reals equipped with this topology
is I-R0 space. It is also clear that (R, τl ) does not form R0 space.
2.3 Let (X, τ ) be a topological space. Then the following statements
are equivalent:
(a) (X, τ ) is an I-R0 space;
(b) foreach x ∈ X and τ -closed set F such that x ∈
/ F, the set of the form
clτ {x} ∩ F belongs to I.
32
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
It is known that if τ and τ1 are topologies in X such that τ ⊂ τ1 and (X, τ )
is R0 space, then (X, τ1 ) need not be R0 space [2]. In the consequence, the same
holds for I-R0 spaces. However, in special cases, it is possible that (X, τ1 ) can
be I-R0 space.
Let (X, τ ) be a topological space and I be an ideal of subsets of X. Let us
assume that
A ∈ I ⇐⇒ A ∩ DI (A) = ∅ ⇐⇒ DI (A) = ∅,
()
where DI (A) is a set of all x ∈ X such that, for each τ -neighborhood V of x,
the set V ∩ A does not belong to I, is true.
We define a new topology τ (I) in X that is introduced by the closure operation
defined in the following way
clτ (I) (A) = A ∪ DI (A).
This topology is called H a s h i m o t o topology [5]. See that we cannot assume
that the space (X, τ ) is T1 space in order the condition () is satisfied (it is
sufficient to consider the real line with the left topology and the ideal of nowhere
dense sets).
2.4 If (X, τ ) is an I-R0 space, then X, τ (I) is an I-R0 space
as well.
P r o o f. Let x ∈ X and F be a τ I -closed
set
such that x ∈
/ F.
in X, τ (I)
It is known that τ ⊂ τ (I), whence clτ (I) {x} ⊂ clτ {x} . On the other hand,
F is closed with respect to the Hashimoto topology, so it can be represented
in the form F = H ∪ N , where H is τ -closed and N ∈ I (because of the
condition ()). By this,
clτ (I) {x} ∩ F ⊂ clτ {x} ∩ F = clτ {x} ∩ (H ∪ N )
= clτ {x} ∩ H ∪ clτ {x} ∩ N .
/ H, the set clτ {x} ∩ H belongs to I.
By assumption
X is I-R0 space and x ∈
Of course, clτ {x} ∩ N ∈ I. Finally, clτ (I) {x} ∩ F is the element of I.
2.5 If (X, τ ) is an I-R0 space then, for each points x, y ∈ X such
that x = y, we have
clτ {x} = clτ {y} or clτ {x} ∩ clτ {y} ∈ I.
P r o o f. Let us assume that x, y ∈ X and x = y. If y ∈
/ clτ {x}
, then by
the second condition
of Theorem
2.3, the
set
clτ {x} ∩ clτ {y} belongs
to
I. If y ∈ clτ {x} , then clτ {x} ⊃ clτ {y} . By symmetry,
/ clτ {y},
if x
∈
then clτ {x} ∩ clτ {y} ∈ I and if x ∈ clτ {y} , then clτ {x} ⊂ clτ {y} .
By those remarks, we get the condition given in our theorem.
33
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
It is known that a topological
space (X,τ ) is R0 space
if and
only
if for each
points x = y we have clτ {x} = clτ {y} or clτ {x} ∩ clτ {y} = ∅ [2], [7].
However, an analogical result cannot be obtained for I-R0 spaces. In other words,
there exists a topological space (X, τ ) and an ideal I of subsets of X such that
(X, τ ) is not I-R0 space but satisfies the property given in Theorem 2.5.
Example 2.2. Let X = {0} ∪ (1, 2) and I = {∅} ∪ A ⊂ (1, 2) : inf A > 1 .
One can show that I forms an ideal of subsets of X. In X, consider
the left
topology restricted to X and denote it by τl|X . Clearly, clτl|X {0} = X and
clτl|X {x} = [x, 2) for x ∈ (1, 2). If we assume that x, y ∈ X and x = y, then
[y, 2) if y > x,
clτl|X {x} ∩ clτl|X {y} =
[x, 2) if x > y.
By that, clτl|X {x} ∩ clτl|X {x} belongs to I. Next, the set {0} is τl|X -open
and 0 ∈ {0}, however. clτl|X {0} \ {0} = (1, 2) ∈
/ I. In the consequence, the
condition in Theorem 2.5 is satisfied but the space is not I-R0 space.
In the next part of this section, we will use the following notations:
(1) If X is a set, J is a family of subsets of X, and M ⊂ X, then JM denotes
a family of subsets of M defined in the following way:
JM = {A ∩ M : A ∈ J} .
(2) If X and Y are nonempty sets and f : X → Y is a function, J is a family of subsets of Y, thenJf will denote a family of subsets of X, where:
Jf = A ⊂ X : f (A) ∈ J .
(3) If X and Y are nonempty sets and f : X → Y is a function, J is a family
f
of subsets
of X, then
by J we define a family of subsets of Y as follows:
f
J = f (A) : A ∈ J .
One can check that:
2.1 If X and Y
are nonempty sets, M ⊂ X and f : X → Y, then:
(a) If I is an ideal of subsets of X, then IM forms an ideal of subsets of M .
(b) If I forms an ideal of subsets of Y, then If forms an ideal of subsets of X.
(c) If I is an ideal of subsets of X, then If is an ideal of subsets of Y.
2.6 If (M, τM ) is a subspace of a topological space (X, τ ) that is
I-R0 space, then (M, τM ) is IM -R0 space.
P r o o f. Let F be a τM -closed in M and x ∈ M \ F . Since F is τM -closed, then
/ F0 . See that
it is of the form F = F0 ∩ M , where F0 is τ -closed. Of course, x ∈
34
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
clτM {x} ∩ F = clτ {x} ∩ M ∩ F0 ∩ M = clτ {x} ∩ F0 ∩ M
and
clτ {x} ∩ F0 ∈ I,
whence clτM {x} ∩ F ∈ IM . In the consequence, M is IM -R0 space.
2.7 Let us assume that:
(a)
(Xs , τs ) s∈S is a family of topological spaces, where Xs ∩ Xs = ∅
for s = s ;
(b) Is is an ideal of subsets of Xs for each s ∈ S;
(c) I is an ideal of X = s∈S Xs such that s∈S Is ⊂ I.
If (Xs , τs ) is Is -R0 space for each s ∈ S, then s∈S Xs is I-R0 space.
P r o o f. Let x ∈ X and F be a τ -closed set that does not contain x. There exists
soF ∩Xs is τs -closed for each
s0 ∈ S such that x ∈ Xs0 . The set F is
τ -closed,
s ∈ S. Let us see that the equality clτ {x} = clτs0 {x} holds. Indeed, the sets
Xs are τ -closed and we get
clτs0 {x} = clτ {x} ∩ Xs0 = clτ {x} .
From the fact that
clτ {x} ∩ F = clτs0 {x} ∩ F ∈ Is0
(1)
and by
assumption that Is0 ⊂ I, we obtain clτ {x} ∩ F ∈ I. We have shown
that s∈S Xs is I-R0 space.
2.8 Let us suppose that:
(a) (Xs , τs )
s∈S
is a family of topological spaces, where Xs ∩Xs = ∅ for s = s ;
(b) Is is an ideal of subsets of Xs for each s ∈ S such that IXs ⊂ Is ;
(c) I is an ideal of X = s∈S Xs such that s∈S Is ⊂ I.
If s∈S Xs is I-R0 space, then (Xs , τs ) is Is -R0 space for each s ∈ S.
P r o o f. If
s∈S Xs ,τ is I-R0 space, then (Xs ,τs ) is a subspace of
s∈S Xs ,τ
and τs = τ|Xs . By Theorem 2.6, it is, IXs -R0 space for each s ∈ S. Clearly, by
the assumption and Theorem 2.6, we get that (Xs , τs ) is Is -R0 space.
2.2 Let (Xs , τs ) s∈S be a family of topological spaces, where
Xs ∩ Xs = ∅ for s = s. If Is is an ideal of subsets of X
s for each s ∈ S
such
that
I
⊂
I
and
I
is
an
ideal
of
subsets
of
X
=
Xs
s
s∈S Xs such that
I
⊂
I,
then
the
following
conditions
are
equivalent:
s∈S s
(a)
s∈S Xs , τ is I-R0 space;
(b) (Xs , τs ) forms Is -R0 space for each s ∈ S.
35
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
2.9 If I is an ideal of subsets of a set X and (X, τs ) is I-R0 space
for each s ∈ S and τ = sup{τs : s ∈ S}, then (X, τ ) is I-R0 space as well.
n
P r o o f. Let x ∈ X and G be a τ -open set. Then, x ∈ i=1 G
si ⊂ G, where
Gsi
is τsi -open set for each i ∈ {1, . . . , n}. Since τsi ⊂ τ , then clτ {x} ⊂ clτsi {x} ,
and consequently,
n n clτ {x} \ Gsi ⊂
clτsi {x} \ Gsi .
clτ {x} \ G ⊂
i=1
By assumption
i=1
clτsi {x} \ Gsi ∈ I,
we have
for all i ∈ {1, . . . , n},
n clτsi {x} \ Gsi ∈ I.
i=1
Finally,
2.3 ([2])
clτ {x} \ G ∈ I.
(a) Every subspace of R0 space is R0 space as well.
(b) A topological sum of R0 spaces is R0 space.
(c) If (X, τs ) is an R0 space for each s ∈ S and τ = sup{τs : s ∈ S},
then (X, τ ) is R0 space as well.
Now, let S be a nonempty set of indices, X = s∈S Xs and ps : X → Xs be
the projection onto the sth factor. If s ∈ S and Is is an ideal of subsets of Xs ,
then the symbol Ips will stand for denotation an ideal of the form
Ips = A ⊂ X : ps (A) ∈ Is .
If I is an ideal of subsets of X then, by Ips, where s ∈ S, we will mean an ideal
of the form
Ips = ps (A) : A ∈ I .
2.10 Let X = s∈S Xs and assume that:
(a) I is an ideal of subsets of X;
(b) Is forms an ideal of subsets of Xs such that Ips ⊂ Is for each s ∈ S.
If (X, τ ) forms I-R0 space then (Xs , τs ) is Is -R0 space for all s ∈ S.
P r o o f. Let s0 ∈ S. Let us take xs0 ∈ Xs0 and let Fs0 be a τs0 -closed set
/ Fs0 . Put Us = Xs , for all s = s0 and Us0 = Fs0 , and denote
such that xs0 ∈
F = s∈S Us . Let x = (xs ) be an arbitrary point from X, where xs has been
chosen for s = s0 in an arbitrary way. It is clear that
clτ {x} ∩ F =
clτs {xs } ∩ Us
and x ∈
/ F,
s∈S
36
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
i.e., by assumption clτ {x} ∩ F ∈ I. Evidently,
ps0 clτ {x} ∩ F = clτs0 {xs0 } ∩ Fs0 ∈ Ips0 ⊂ Is0 .
Since s0 has been chosen in an arbitrary way, we get that (Xs , τs ) forms Is -R0
space for each s ∈ S.
2.11 Let Is form an ideal
of subsets of Xs and I be an ideal of
subsets of X = s∈S Xs such that s∈S Ips ⊂ I. If (Xs , τs ) is Is -R0 space for
each s ∈ S then (X, τ ) is I-R0 space.
P r o o f. Let x = (xs ) ∈ X, F be a τ -closed set in X and assume that x ∈
/ F.
By
this,
x
∈
X
\
F
,
i.e.,
there
exists
a
family
of
sets
{W
}
such
that
x ∈
s s∈S
s∈S Ws ⊂ X \ F , Ws is open in Xs for each s ∈ S and Ws = Xs for finite
quantity of s ∈ S. Let S0 ⊂ S be a set of all indexes such that Ws = Xs and let
us say S0 = {s1 , . . . , sn }. Put
Fsi ,
where i ∈ {1, 2, . . . , n},
Fi =
s∈S
Fsi
= Xs for s = si and Fsii = Xsi \ Wsi for i ∈ {1, 2, . . . , n}. Then,
F ⊂ F1 ∪ . . . ∪ Fn
and
x∈
/ F1 ∪ . . . ∪ Fn .
Let us see that
psi clτ {x} ∩ Fi = clτsi {xsi } ∩ (Xsi \ Wsi ) ∈ Isi ,
/ Xsi \Wsi and (Xsi , τsi ) is Isi -R0 space, i.e., clτ {x} ∩Fi ∈ Ipsi ⊂ I.
since xsi ∈
As a result, we have
n
n clτ {x} ∩ Fi ∈ I,
Fi =
clτ {x} ∩ F ⊂ clτ {x} ∩
i=1
i=1
so
clτ {x} ∩ F ∈ I.
Consequently, (X, τ ) is I-R0 space and the proof is completed.
The next theorem will concern the inverse system of topological spaces (for details, see [3]). The denotations are those used in this book.
2.12 Let (Xσ , τσ ) be an Iσ -R0 space for each σ ∈ Σ, S = {Xσ , πσ , Σ}
be an inverse system, and let I be an ideal such that σ∈Σ Ipσ ⊂ I. If X0 denotes
the limit of the inverse system S, then X0 forms IX0 -R0 space.
P
r o o f. By
the previous theorem, the Cartesian product of topological spaces
(Xσ , τσ ) σ∈Σ is I-R0 space. The limit of the inverse system forms a subspace
of the Cartesian product, whence it is IX0 − R0 space by Theorem 2.6.
37
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
2.4 ([2])
(a) The Cartesian product s∈S Xs is an R0 space if and only if (Xs , τs ) is
an R0 space for each s ∈ S.
(b) If (Xσ , τσ ) is an R0 space for each σ ∈ Σ and S = {Xσ , πσ , Σ} is an
inverse system, then the limit of the inverse system S is an R0 space.
3. I-T3 spaces
We here introduce the concept of I-T3 spaces useful in the next section. In the
beginning, we will give connections between such a kind of spaces and spaces
that satisfy the well-known axioms of separation. The last theorem of this section
will describe the operations for I-T3 spaces.
3.1 Let I be an ideal of subsets of a set X. We will say that
a topological space (X, τ ) is I-T3 space if, for each x ∈ X and τ -closed set F
such that x ∈
/ F , there exists a τ -open set V such that x ∈ V and clτ (V )∩F ∈ I.
Remark 3.1 It is not difficult to see that a topological space (X, τ ) is I-T3
space if and only if, for each x ∈ X and each G τ -open set such that x ∈ G,
there exists a τ -open set V such that x ∈ V with the property clτ (V ) \ G ∈ I.
3.2 Every T3 space is I-T3 space.
P r o o f. Let (X, τ ) be a T3 space. Take then x ∈ X and a τ -closed set F such
that x ∈
/ F . Then, X \ F is τ -open and x ∈ X \ F . By assumption, X is T3
space, whence we can find a τ -open set V such that x ∈ V ⊂ clτ (V ) ⊂ X \ F.
By this, clτ (V ) ∩ F = ∅ ∈ I and V is a τ -neighborhood of x. It means that that
(X, τ ) is I-T3 space.
There exists an I-T3 space that is not a T3 space.
Example 3.1. It is sufficient to take into account the space of reals equipped
with the left topology τl and the ideal of τl -nowhere dense sets.
3.3 Every I-T3 space is I-R0 space.
P r o o f. Let (X, τ ) be an I-T3 space and choose x ∈ X and a τ -closed set F
such that x ∈
/ F . By assumption, X is I-T3 space, i.e.,
there
exists a τ -open set
V such that x ∈ V and
cl(V
)
∩
F
∈
I.
Moreover,
cl
{x}
∩ F ⊂ cl(V ) ∩ F and
consequently, cl {x} ∩ F ∈ I.
There exists an I-R0 space that is not I-T3 space.
38
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
Example 3.2. In the space R, let us define the topology
τ = {A ⊂ R : R \ A is countable} ∪ {∅}.
We declare that A ∈ I if and only if it is finite subset of R. Taking x ∈ R and
V a τ -neighborhood of x, we get clτ {x} = {x} ⊂ V . Hence, (R, τ ) is R0 space
and, in the consequence, I-R0 space. Now, let x ∈ IQ and F = Q. The set F
is τ -closed and x ∈
/ F . If V is a τ -neighborhood of x then clτ (V ) = R which
implies clτ (V ) ∩ F = R ∩ F = F ∈
/ I. It means that (R, τ ) is not an I-T3 space.
By the above example, there exists R0 space that is not I-T3 space. If we consider
the space of reals with left topology and ideal of nowhere dense sets with respect
to the topology, then it is I-T3 space that is not R0 space. In the consequence,
there is no connection between I-T3 spaces and R0 spaces. In general, we have
the following diagram
T3 space
⏐
⏐
−−−−→
R0 space
⏐
⏐
I − T3 space −−−−→ I − R0 space
3.4 The following statements hold:
(a) If (M, τM ) is a subspace of a topological space (X, τ ) that is I-T3 space,
then (M, τM ) is IM -T3 space.
(b) Let (Xs , τs ) s∈S be a family of topological spaces, where Xs ∩ Xs = ∅ for
s = s . Let
Is be an ideal ofsubsets of Xs for each s ∈ S and I be an ideal
of X = s∈SXs such that s∈S Is ⊂ I. If (Xs , τs ) is Is -T3 space for each
s ∈ S, then s∈S Xs is I-T3 space as well.
(c) Let (Xs , τs ) s∈S be a family of topological spaces, where Xs ∩ Xs = ∅
for s = s. Assume that Isis an ideal of subsets
of Xs such that
IXs ⊂ Is
and I is an ideal of X = s∈S Xs such that s∈S Is ⊂ I. If s∈S Xs is
I-T3 space, then (Xs , τs ) is Is -T3 for each s ∈ S.
(d) If (X, τs ) is I-T3 space for each s ∈ S and τ = sup{τs : s ∈ S}, then (X, τ )
is I-T3 space as well.
(e) Let X = s∈S Xs . Assume
(1) I is an ideal of subsets of X;
(2) Is form an ideal of subsets of Xs such that Ips ⊂ Is for each s ∈ S.
If (X, τ ) forms I-T3 space, then (Xs , τs ) is Is -T3 space for all s ∈ S.
(f) Let I
s forms an ideal ofsubsets of Xs and I be an ideal of subsets of
X = s∈S Xs such that s∈S Ips ⊂ I. If (Xs , τs ) is Is -T3 space for each
s ∈ S, then (X, τ ) is I-T3 space.
39
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
(g) Let (Xσ , τσ ) be an Iσ -T3 space for each σ ∈ Σ, S = {Xσ , πσ , Σ} be an
inverse system and I be an ideal such that σ∈Σ Ipσ ⊂ I. If X0 denotes
the limit of the inverse system S, then X0 forms IX0 -T3 space.
P r o o f. (a) Let x ∈ M and let F be a τM -closed set such that x ∈ M \ F .
Since F is τM -closed then it is of the form F = F0 ∩ M , where F0 is τ -closed
and x ∈
/ F0 . By assumption, there exists a τ -open set V0 such that x ∈ V0 and
clτ (V0 ) ∩ F0 ∈ I. Let us put V = V0 ∩ M . Then V ∈ τM , x ∈ V and
clτM (V ) ∩ F ⊂ clτ (V0 ) ∩ F0 ∩ M ∈ IM ,
so, clτM (V ) ∩ F ∈ IM . It means that (M, τM ) is IM -T3 space as well.
(b) Let us assume that x ∈ X, F is τ -closed, and x ∈
/ F. There exists exactly
/ F ∩ Xs0 .
one s0 ∈ S such that x ∈ Xs0 . Of course, F ∩ Xs0 is τs0 -closed and x ∈
By assumption, there exists a τs0 -open set V ⊂ Xs0 such that x ∈ V and
clτs0 (V ) ∩ F ∈ IXs0. Then, V is also τ -open, x ∈ V, and
clτ (V ) ∩ F = clτs0 (V ) ∩ F ∈ Is ⊂ I,
completes the proof.
(c) Since (Xs , τs ) forms a subspace of the considered topological space, it is
IXs -T3 space, and so, it is Is -T3 space.
(d) Let x ∈ X and let F be a τ -closed set suchthat x ∈
/ F. Then,there exist
n
n
/ i=1 Fi . By
τi -closed sets Fi for i = 1, . . . , n such that F ⊂ i=1 Fi and x ∈
assumption, one can find a τi -open set Vi such that x ∈ Vi and clτi (Vi ) ∩ Fi ∈ I
n
for i = 1, . . . , n. Let V = i=1 Vi . Of course, V is τ -open set, x ∈ V and
clτ (V ) ∩ F ⊂
n
clτi (Vi ) ∩
i=1
n
i=1
Fi ⊂
n
(clτi (Vi ) ∩ Fi ) ∈ I,
i=1
i.e., clτ (V ) ∩ F ∈ I and the proof is finished.
(e) Let us take s0 ∈ S, xs0 ∈ Xs0 , and let Fs0 be a closed set in X
s0 such
/ Fs0 . Denote Us0 = Fs0 and Us = Xs for s = s0 . Of course, s∈S Us
that xs0 ∈
is closed in X, x = (xs ) ∈
/ s∈S Us , where xs is an arbitrary fixed point from
Xs for s = s0 .By assumption, there exists an open set V in X such that
x∈V
and clτ (V ) ∩ s∈S Us ∈ I. One can find an open set W of the form s∈S Ws ,
where Ws = Xs for a finite quantity of indexes s ∈ S and x ∈ W ⊂ V. By the
reason that xs0 ∈ Ws0 and
ps0 clτ (W ) ∩
Us = clτs0 (Ws0 ) ∩ Fs0 ∈ Ips0 ⊂ Is0 ,
s∈S
we get that (Xs0 , τs0 ) forms Is0 − T3 space.
/ F.
(f) Let x = (xs ) ∈ X and let F be a closed subset of X such that x ∈
Then, x ∈ X \ F and X \ F is open in X. By this, we can find a family of sets
40
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
{Ws }s∈S such that Ws is open in Xs for
each s ∈ S, Ws = Xs for a finite family
of indexes S0 = {s1 , . . . , sn } ⊂ S, x ∈ s∈S Ws ⊂ X \ F. Put
Fsi ,
Fi =
s∈S
Fsi
Fsii
= Xs for s ∈
/ S0 and
= Xsi \ Wsi for i ∈ {1, . . . , n}. From that,
where
F ⊂ F1 ∪ . . . ∪ Fn and x ∈
/ F1 ∪ . . . ∪ Fn . By assumption, there exists
a τsi -open
set Vsi such that xsi ∈ Vsi and clτsi (Vsi ) ∩ Fsii ∈ Isi . Let Vi = s∈S Vsi , where
Vsi = Xs for s = si . Whence
psi cl(Vi ) ∩ Fi = clτsi (Vsii ) ∩ Fsii ∈ Isi .
It means that cl(Vi ) ∩ Fi belongs to Ipsi and, in the consequence, to I for
n
i ∈ {1, . . . , n}. Finally, let V = i=1 Vi . Clearly, x ∈ V, V is an open subset
in X, and
clτ (V ) ∩ F ⊂
n
i=1
n
clτsi (V ) ∩ Fi ⊂
clτsi (Vi ) ∩ Fi ∈ I.
i=1
As a result, we get cl(V ) ∩ F ∈ I. It has been shown that X forms I-T3 space.
(g) It is a consequence of (a) and (f).
3.1
(a) A subspace of a topological space (X, τ ) that is T3 space is T3 space.
(b) Let (Xs , τs ) s∈S be a family of topological spaces, where Xs ∩ Xs = ∅
for s = s.
s∈S Xs is a T3 space if and only if (Xs , τs ) is a T3 space
for each s ∈ S.
(c) If (X, τs ) is a T3 space for each s ∈ S and τ = sup{τs : s ∈ S}, then (X, τ )
is a T3 space as well.
(d) s∈S Xs is a T3 space if and only if (Xs , τs ) is a T3 space for each s ∈ S.
(e) If (Xσ , τσ ) is a T3 space for each σ ∈ Σ and S = {Xσ , πσ , Σ} is an inverse
system, then the limit of the inverse system S is a T3 space.
4. I-strong quasi-uniform convergence
In this section, we define a new kind of convergence of sequences of function.
We will investigate the connections between this kind of convergence and the
well-known kinds of convergence of sequences of functions, in particular, the
pointwise convergence. In the end, we formulate some more general results to
those ones contained in [9] and [10].
41
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
4.1 Let I be an ideal of subsets of a nonempty set X. A family of
s ∈ S} of a topological space (X, τ ) is said to be an I-cover of X
sets α = {Us : if the set X \ s∈S Us belongs to I.
Moreover, if {Us : s ∈ S} is I-cover that consists of open sets, then such an
I-cover is said to be an open I-cover of X.
4.2 We willsay that an open I-cover α of a topological space X
contains a point x if x ∈ U ∈α U .
4.3
Assume that (X, τX ) and (Y, τY ) are topological spaces and
f, fn : X → Y are functions for n ∈ N. We say that a net (fn )∞
n=1 is I-strongly
quasi-uniformly convergent to f if, for each x0 ∈ X and for each open I-cover
α of Y that contains f (x0 ), there exists n0 ∈ N such that if n ≥ n0 then
there
exists a neighborhood U of x0 such that α contains f (x) and fn (x) ∈ St f (x), α
for x ∈ U.
4.1 If (X, τX ) and (Y, τY ) are topological spaces, f, fn : X → Y are
functions and (fn )∞
n=1 is I-strongly quasi-uniformly convergent to f, then it is
strongly quasi-uniformly convergent to f.
P r o o f. It follows from the fact that each open cover of the space Y is an open
I-cover of Y.
Example 4.1. Let X = Y = R be equipped with a left topology τl . Consider
an arbitrary sequence (fn )∞
n=1 of functions and a function f such that fn , f are
defined on X with values
in
Y. If α is an open cover of Y then, for each x ∈ R,
we have fn (x) ∈ St f (x), α = R, whence (fn )∞
n=1 is strongly quasi-uniformly
convergent to the function f.
Now, let us consider the functions fn (x) = −x + n, where x ∈ R and n ∈ N.
Let I be an ideal of τl -nowhere dense sets, x0 = 0, and take
into account
a function f : R → R such that f (−1) ∈ (−∞, 1). Let α = (−∞, 1) and
n ∈ N. Of course, α forms open I-cover of R. Furthermore, one can find n0 ∈ N
such that for each τl -neighborhood
U of x0 , we have fn0 (−1) > 1. By this,
/ St f (−1), α = (−∞, 1). Hence, (fn )∞
fn0 (−1) ∈
n=1 does not I-strongly quasi-uniformly converge to f.
4.2 Assume that (X, τX ) and (Y, τY ) are topological spaces. Let
f, fn : X → Y for n ∈ N. Let I1 and I2 be ideals of a subset of Y such that
∞
I1 ⊂ I2 . If (fn )∞
n=1 is I2 -strongly quasi-uniformly convergent to f, then (fn )n=1
is I1 -strongly quasi-uniformly convergent to f.
P r o o f. It follows from the fact that each open I1 -cover is an open I2 -cover. Remark, if a sequence (fn )∞
n=1 is I1 -strongly quasi-uniformly convergent to f,
then it need not be I2 -strongly quasi-uniformly convergent to f (see Example 4.1).
42
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
4.3 Assume that (X, τX ) and (Y, τY ) are topological spaces and I is
an ideal of subsets of Y. If fn , f : X → Y for each n ∈ N, (Y, τY ) is an I-R0 space
and (fn )∞
n=1 is I-strongly quasi-uniformly convergent to f, then it is pointwise
convergent to f.
P r o o f. Let x0 ∈ X and U be a τY -neighborhood
of f (x0 ). By assumption,
the set
cl
)}
\
U
belongs
to
I.
Let
α
=
U,
Y
\
cl
({f
(x
)})
. Then
{f
(x
τ
0
τ
0
Y
Y
Y \ U ∪ (Y \ clτY ({f (x0 )})) = clτY {f (x0 )} \ U ∈ I. By this, α is an open
I-cover of Y and f (x0 ) belongs to α. From the assumption, there exists n0
such that if n ≥ n0 , then one can find a τY -neighborhood U of
x0 such
that
fn (x) ∈ St f (x), α for each x ∈ U . In the sequel, fn (x0 ) ∈ St f (x0 ), α = U,
i.e., fn (x0 ) ∈ U for n ≥ n0 and the proof is completed.
4.1 ([10]) Let f : X → Y, fn : X → Y, and let Y be an R0 -space.
is
strongly
quasi-uniformly convergent to f, then it is pointwise conIf (fn )∞
n=1
vergent to f.
P r o o f. It is sufficient to take into account the trivial ideal I = {∅}.
Let us remark that the condition in Theorem 4.3 “(fn )∞
n=1 is I-strongly quasi-uniformly convergent to the function f ” cannot be replaced with the weaker condition “(fn )∞
n=1 is strongly quasi-uniformly convergent to the function f ”(see Example 4.1).
4.4 Let (fn)∞n=1 be a net of functions defined on a topological space
(X, τX ) with values in a T3 space (Y, τY ) and f : X → Y be a continuous function. Then the following conditions are equivalent:
(a) (fn )∞
n=1 is strongly quasi-uniformly convergent to a function f ;
(b) (fn )∞
n=1 is I-strongly quasi-uniformly convergent to a function f for each
ideal I of subsets of Y.
P r o o f. Let (fn )∞
n=1 be a strongly quasi-uniformly convergent net to a function f. Let x0 ∈ X and α be an open I-cover of Y that contains f (x0 ). There
exists U ∈ α such that f (x0 ) ∈ U , whence one can
find a τY -open
set V such
that f (x0 ) ∈ V ⊂ clτY (V ) ⊂ U . Let us put α = U, Y \ clτY (V ) . The family
α forms an open cover of Y. An I-strongly quasi-uniform convergence implies
an existence of n0 ∈ N such that
for n ≥ n0 there is a τX -neighborhood W of x0
with the property fn (x) ∈ St f (x), α for each x ∈ W . By continuity of f, there
exists a τX -neighborhood W of x0 such that f (W ) ⊂ V. The set W0 = W ∩ W is τX -open and x0 ∈ W0 . Then, if n ≥ n0 , we have
fn (x) ∈ St f (x), α = U ⊂ St f (x), α
for each x ∈ W0 . It means that (fn )∞
n=1 is I-strongly quasi-uniformly convergent
to f.
43
ROBERT DROZDOWSKI — AGATA SOCHACZEWSKA
The implication from (b) to (a) follows from Theorem 4.1.
4.5 If (X, τX ), (Y, τY ) are topological spaces, I forms an ideal of sub-
sets of Y, fn , f : X → Y are continuous for each n ∈ N, and (fn )∞
n=1 is pointwise
convergent to f, then it is I-strongly quasi-uniformly convergent to f.
P r o o f. The proof of this theorem is analogous to that of Theorem 3 in [9],
so it is omitted here.
4.2
If (X, τX ), (Y, τY ) are topological spaces, fn , f : X → Y are
continuous for each n ∈ N, and (fn )∞
n=1 is pointwise convergent to f, then it is
strongly quasi-uniformly convergent to f.
4.3 Let fn , f are continuous functions defined on a topological
space X with values in a T3 space Y for every n ∈ N. Then the following conditions are equivalent:
(1) (fn )∞
n=1 is pointwise convergent to f ;
(2) (fn )∞
n=1 is strongly quasi-uniformly convergent to f ;
(3) (fn )∞
n=1 is I-strongly quasi-uniformly convergent to f for each ideal I of
subsets of Y.
4.6 Let (X, τX ) be an arbitrary topological space and (Y, τY ) be
a topological space that is an I-T3 space, where I is an ideal of subsets of Y.
If (fn )∞
n=1 is a net of functions such that fn : X → Y is continuous for each
n ∈ N and (fn )∞
n=1 is I-strongly quasi-uniformly convergent to a function f,
then f : X → Y is continuous.
P r o o f. Let x0 ∈ X and W be a τY -neighborhood of f (x0 ). There exists a τY -open set V such that f (x0 ) ∈ V and clτY (V )∩(Y \W ) = H ∈ I. Let us consider
the family α = {W, Y \ clτY (V )}. The family α forms an open I-cover of Y.
Since Y is I-T3 space then it is I−R0 space. By Theorem 4.3, (fn )∞
n=1 is pointwise
convergent, i.e., there exists n0 such that fn (x0 ) ∈ V for all n ≥ n0 . On the other
U
hand, there exists n1 such that for n ≥ n1 one can find a τX -neighborhood
of x0 with the properties: α contains f (x) and fn (x) ∈ St f (x), α for all x ∈ U .
Now, let us choose m ≥ max{n0 , n1 }. Since fm is continuous then there exists a
τX -neighborhood U of x0 such that fm (x) ∈ V for x ∈ U .Let U0 =
U ∩U . U0
is τX -neighborhood
of x0 and
fm (x) ∈ V and fm (x) ∈ St f (x), α for x ∈ U0 .
By this, f (x) ∈ St fm (x), α = W for each x ∈ U0 . It means that f : X → Y is
continuous and the proof is completed.
4.4 ([9])∞ Let (X, τX ) be an arbitrary topological space
and (Y, τY )
∞
be a T3 space. If (fn )n=1 is a net of continuous functions and (fn )n=1 is strongly
quasi-uniformly convergent to a function f, then f is continuous.
44
ON THE I-R0 SPACES AND I-STRONG QUASI-UNIFORM CONVERGENCE
REFERENCES
[1] ARZELÀ, C.: Sulle serie di funzioni, Mem. R. Accad. Sci. Inst. Bolonga Ser. 5(8)
(1899–1900), 130–186, 701–744.
[2] CSÁSZÁR, Á.: General Topology. Akadémiai Kiadó, Budapest, 1978.
[3] ENGELKING, R.: General Topology. Warsaw, 2007. (In Polish)
[4] EWERT, J.: On strong form of Arzela convergence, Intemat. J. Math. Sci. 20 (1997),
417–422.
[5] HASHIMOTO, H.: On the -topology and its application, Fund. Math. 91 (1976), 5–10.
[6] KUPKA, I.—TOMA, V.: A uniform convergence for non-uniform spaces, Publ. Math.
Debrecen 47 (1995), 299–309.
[7] MURDESHWAR, M. G.—NAIMPALLY, S. A.: Quasi-Uniform Topological Spaces.
Noordhoff, Groningen, 1966.
[8] PREDOI, M.: Sur la convergence quasi-uniform, Period. Math. Hungar. 10 (1979), 31–40.
[9] SOCHACZEWSKA, A.: The strong quasi-uniform convergence, Math. Montisnigri 15
(2002), 45–55.
[10] SOCHACZEWSKA, A.: Strong quasi-uniform convergence. in: Reports on Real Analysis,
Conference at Rowy, 2003, pp. 175–178.
Received April 9, 2011
Institute of Mathematics
Academia Pomeraniensis
Arciszewskiego 22d
76-200 Slupsk
POLAND
E-mail : [email protected]
agata [email protected]
45