Comparison of Several Topologies Generated by the Convergence
DORIS DODA1, ELFRIDA DISHMEMA2, AGRON TATO3, JONIDA KAPXHIU4,
MOANA GOREZI5
1Department of Economy, Entrepreneurship, and Finance, Barleti University,
Frang Bardhi Street, Tirana, ALBANIA
2 Department of Mathematics and Informatics, Agriculture University of Tirana, Koder Kamez,
Tirana, ALBANIA
3 Department of Mathematical Engineering, Polytechnic University of Tirana,
Sulejman Delvina Street, Tirana, ALBANIA
4 Department of Economy Wisdom, University College,
Fuat Toptani Street, Tirana, ALBANIA
5Turgut Ozal College, Tirana-Durres Highway,
Tirana, ALBANIA
Abstract: This paper studies the construction of three functional topologies in production spaces. Their
topological bases have been found and compared between them. An important place is a comparison with well-
known topologies such as uniform, point, and open compact convergence. The convergences used are uniform
local convergence, strongly uniformed local convergence, and the convergence known as -convergence.
Keywords. Generation of new topologies, locally uniform convergence, convergence locally uniformly
strongly, -convergence, closure operator.
Received: October 12, 2022. Revised: December 14, 2022. Accepted: January 11, 2023. Published: February 2, 2023.
1 Introduction
Let's briefly present some of the concepts we used
to carry out this study.
Definitions 1.1: [1] Let (X, d), (Y, p) be two metric
spaces xX, and
,:
n
f f X Y
. The function
is a δ-limit of the sequence if for
every ε>0, there exists
and that
for every , and , we have
.
Thus, we can say that
converges
locally uniformly to f(x).
(2). Let (X, d), (Y, p) be two metric spaces, xX,
and
,:
n
f f X Y
. We say that the sequence
is locally uniformly strongly convergent (or
short
a -convergent) to f(x) if for every >0 and xX,
there exists
and >0, such that for
0( , )n n x
and
( , )y S x
we have
( ( ), ( ))
n
p f y f x
(3) It is said that the sequence of functions
is -convergent in X if:
then This means that
when
then
, [3].
Thus we can say that converges locally
uniformly to f(x), [1].
Definition 1.2. The closure operator is called
the operator that enjoys the following
properties:
=
=.
Let X be any set, by generating a topology on X we
mean the selection of the family τ of subsets of X
that satisfies the known conditions of open sets. It is
often more useful not to describe the family τ of
open sets directly. The other methods consist first of
all in defining a family that serves as the basis of the
topology, or of the adjacency system, of the closure
operator or the interiority operator. In this paper,
we are considering the closure operator.
Suppose that given a set X and closure operator,
each AX defines a set X such that
conditions (C01) -(C04) are satisfied. The
family τ= {X|A: A=} satisfies the conditions
of open sets and for each AX the set is the
closure of A in the topological space (X, τ). The
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DOI: 10.37394/23206.2023.22.13
Doris Doda, Elfrida Dishmema,
Agron Tato, Jonida Kapxhiu, Moana Gorezi
E-ISSN: 2224-2880
109
Volume 22, 2023
topology τ is called the topology generated by
the closure operator.
2 Main Results
Let us carry out in this section the comparison of
some uniform Cartesian production topologies.
Let X and Y be two topological spaces
corresponding to the product of all continuous
functions leading X to Y. We denote by F a family
of functions contained in and we raise the
question of whether there are topologies in
belonging to the family F. Let's examine the
mechanism of construction of these topologies
related to the family of functions.
Let A and f where X, Y be two metric
spaces. We define these convergences:
(1)
if , where , for i=1,2,
…,
With the equation , we understand that
the sequence is uniformly convergent to f.
(2) Let x X and V be a neighborhood of f(x).
Denote ) and , then if
where , for i=1, 2, … and with
we understand uniform convergence to f
for every .
(3) Let and be the neighborhood of
. We denote , where
()
D
fx
EV
,
if , , and
with , we understand that
()
( ( ), ( ))
i f x
f x f x V
for every .
(4) Let and be the neighborhood of
. We denote , where
()
F
fx
GV
if , , and with for
i=1, 2, …, we understand that
()
( ( ), ( ))
i i f x
f x f x V
for every .
Proposition 2.1. The closure operator defined
on by each of the formulas (1), (2), (3), (4)
satisfies the conditions (C1), (C2), (C3), (C4)
and defines the respective topology.
Proof. We do the proof only for (2) since they are
similar for other operators (for (1) see, [2]).
Condition (C1) is fulfilled. Since when ,
i=1, 2, … it follows that it turns out
that the condition (C2) is also fulfilled.
It immediately follows from (2) that
when then
,
since, to prove condition (C3), it suffices to
show that
(iii)
.
Taking
it follows that there exists the
sequence of functions belonging to AB and
that which means for or in B and
every
D
CV
, where V is a neighborhood of f,
there is δ>0, that for x D and ,
or that This shows that (C3) is
satisfied.
It follows from statement (ii) that
.
In order to prove the condition (C4), it is sufficient
to prove that
Taking
the sequence will belong to
and satisfy the equation which means that
if for every , there is , and
such that for
and , ,
from which it follows that . But for this
function the sequence
such that
, where
and
D
CV
for j=1, 2, ... g and here it will be found such
that
,
Since the functions
belong to C for k=1, 2, 3…
from (v) and (vi) it follows that which
completes the proof.
(I) Following, [2], the topology generated by
operator (1) is called the uniform convergence
topology on . It can be easily verified that for a
, the family
, where
is a basis at f of the space with the topology of
uniform convergence.
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For the topologies introduced in this paper we will
have:
II) The topology generated by the operator (2) is
called locally uniform convergence topology in .
For each
family
, where
,
is a basis at f for the space endowed with the
locally uniform topology.
(III) The topology generated by operator (3) is
called the topology of -convergence in . Here
too for each , family
, where
The topology generated by operator (4), is called the
topology of -convergence in . Here too for each
f , family
, where
This family is a basis at f for the space equipped
with the -topology. Considering the topology in a
subspace, we conclude e.g., that the space of
uniform convergence in induces the uniform
topology of the subspace in where I is the
unit interval.
Regarding this subspace, the statement has been
proved:
Proposition 2.2. For any topological space X, the
set is closed in the space with uniform
convergence topology, [2].
Let X and Y be two arbitrary topological spaces for
which AX and BY.
We determine
(vii) .
Denote the family of finite subsets of and
denote τ the topology on Y. The family β of all
sets
, where and
for i=1, 2, ..., k generates, as is known, [2], [3] a
topology in which is called the topology of
pointwise convergence in .
Due to the proof below, we bring to attention,
together with the argument, a well-known statement
in topology following in this case [2].
Proposition 2.3. The topology of pointwise
convergence in coincides with the topology of
the Cartesian product subspace where
for every
Proof. As is known, any open space in ,
equipped with the topology of the Cartesian product
subspace is a union of sets in the form
(viii)
where and for i=1, 2, …, k.
But considering that
(ix)
it turns out that the sets of the form (viii) and all the
sets that are open concerning the topology of the
Cartesian product subspace are open concerning the
pointwise convergence topology.
Conversely, from equation it follows that for
and we will have
which means that all sets that are open concerning
the pointwise convergence topology are also open
concerning the topology of a subspace of the
Cartesian product, [2].
Corollary 2.4. A net in the space
with pointwise convergence topology converges
to if and only if the net }
converges to f(x) for every x X, [1].
If we compare the topologies constructed from
definitions (1), (2), (3), and (4) it is not difficult to
establish a ranking. Since for each C (of
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convergences (2) and (3)), there exists an A such
that then the topology of uniform
convergence is richer than the topology of locally
uniform convergence. Likewise, since for every C,
there is a D such that
, it turns out that the
topology of -convergence is richer than the
topology of locally uniform convergence by one
more type,
FD
, so, we have proved the
assertion.
Theorem 2.5.: ,
where
,lu
is the locally uniform convergence
topology,
a
is the convergence’s topology, is
the -convergence topology, and
u
is the uniform
convergence topology.
The above inclusions can be given in the metric
spaces also using the corresponding spheres:
(1) [B,, ] l. u. = {(f, g): ,; p(f(y), g(y)) < ,
yS(x,)}
(2) [B,, ]a = {(f, g): ,; p(f(x), g(y)) < ,
yS(x,)}
(3) [B,, ] = {(f, g): ,; p(f(x), g(xi)) < ,
xiS(x,)}
(4) [B,]u = {(f. g): p(f(x), g(x)) < , x X}
Knowing that the topology of uniform convergence
is richer than the topology of pointwise
convergence, the question arises: If the other two
convergences shown above are between these two
topologies?
Proposition 2.6. For any topological space X, the
locally uniform convergence topology on is
richer than the pointwise convergence topology.
Proof: The equivalence of conditions (i) and (v) in
proposition 1.4.1, [2], it is shown that it suffices to
prove that when is in the closure of the set
, concerning the locally uniform
convergence topology then f is in the closure of A
concerning the pointwise convergence topology.
Let
be a neighborhood of f in the pointwise convergence
topology of Proposition 2.3. As long as are
opened in there is an , such that
, for
As long as f is continuous
is opened and Since,
, it follows that
for j=1, 2,.., n
and every This shows that
.It follows that .
In many manuals of topology and functional
analysis such as, [2], [3], related to topologies in the
space of continuous functions , where X and Y
are topological spaces, a prominent place is the
study of topological spaces of pointwise
convergence and also of compact open topology.
Following [4], it is proved that the compact-open
topology is richer than the topology of pointwise
convergence. Let us compare below an independent
way of how the topology of locally uniform
convergence is related to the compact open-
convergence. The compact-open topology in is a
topology generated from the basis consisting of the
sets
, where is a compact set in X
and is an open set in Y for i=1, 2, ..., k. In
general, the compact-open topology in differs
from the topology of uniform convergence,
however, in the case of it is observed that this
topology coincides with the topology of pointwise
convergence.
Proposition 2.7. For any topological space X, the
locally uniform convergence topology on is
richer than the compact open topology.
Proof: Even here it suffices to prove that when
is in the closure of the set C considered in
(2), where
concerning the locally uniform
topology then f is in the closure A in relation to
compact open topology.
As we noted, the basis of the open compact
topology has the form
1( , )
k
ii
iM K U
,
where, Ki are compact sets in X, whereas Ui are
opened sets in and
M (Ki, Ui) = {f: f(Ki)Ui),
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Let
, be a neighbourhood
of f in the compact open topology, and
opened in , then there exists , such that
for i=1, 2,…, k. Given
the definition of locally uniform convergence
( ) lim ( )
j
j
f x f x
, for every > 0 and for xX,
there exists x and n (x, ), such that for xi
S(x,), it follows that |fj(x)-f(x)| < .
Since f(xi) Ui it follows that xi Ki. From the fact
that Ki is a compact set from any covering
{S (x, x)}, any covering of it will yield a finite sub-
covering.
If xi will take part in one of them, e.g., xi
( , )
j
jx
Sx
then |fj(xi)-f(xi)| <. This means that fj(xi)
Ui from which it follows that U A .
3 Conclusion
It was proved that the α topology is richer than the
topology, richer than the locally uniform
topology, and that the locally uniform topology is
richer than the compact-open topology.
References:
[1] Doda, D., Exhaustive nets on function spaces,
Wseas Transactions in Mathematics, 22
(2023) p. 42-46.
https://doi.org/10.37394/23206.2023.22.5
[2] Engelking, R., General topology, Berlin,
Helderman, 1989.
https://zbmath.org/0684.54001
[3] Gregoriades, V., Papanastassiou, N., The
notion of exhaustiveness and Ascoli-type
theorems, Topology and its applications,155
(2008) p. 1111-1128.
https://doi.org/10.1016/j.topol.2008.02.005
[4] Kelley, J., General topology, Springer-Verlag,
1975.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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