Exhaustive Nets on Function Spaces
DORIS DODA
Department of Economy, Entrepreneurship, and Finance
Barleti University
Frang Bardhi Street
ALBANIA
Abstract: The observation of α-convergence is done in many publications in different contexts and tools.
Recently, we defined the da-convergence in metric space and we emphasized that in every case d -converge or
so-called locally uniform convergence opens new ways for studying relationships in different spaces. In this
paper, we extend the convergence of sequences to the net's convergence and arrived to relativize some known
propositions.
Keywords: exhaustiveness, exhaustive sequence, exhaustive net, d-converge, da-convergence
Received: October 31, 2022. Revised: November 5, 2022. Accepted: December 2, 2022. Available online: December 22, 2022.
1- Exhaustive Functions
We now introduce a new notion of exhaustiveness
close to the idea of equicontinuity, [1].
Definition 1.1 Let (X, d), (Y, p) be metric spaces,
x X, and be a family of functions from X to Y.
Let , be the functional sequence.
(1) If is infinite, we call this family exhaustive at
xX if for each ε > 0 there exists δ> 0 and any finite
set A as a subset of such that: for each y S (x, )
and for each f \A we have that
(2) In the case where is finite we define that is
exhaustive at x if each member of , is a continuous
function at X.
(3) is called exhaustive if is exhaustive at every
xX.
(4) The sequence is called exhaustive at x if
for all ε> 0 there exist δ >0 and n0 , such that for
every yS(x,) and every n>n0 we have
that
(5) The sequence i s called exhaustive if it is
exhaustive at every x X, [1].
Notice that in the most interesting case
where is a sequence of functions such that
for mn we have f nf m for which, then the family
is exhaustive at some point x, if and only if the
sequence is is exhaustive at point x.
Part of this paper will be a special convergence
commonly called α- convergence.
It was originally called continuous convergence and
has been known since the beginning of the 20th
century. This concept, from the literature, written in
the 1950 year we knew that [2] and [5], bring some
facts about it, and by them, it is possible to
characterize this type of convergence, [2], [5].
Expressed non-rigorously, α-convergence is
characterized by two convergences. If x n x, then f
(x n)f(x). It is understood that when function f
maps two spaces, f: (X, d)(Y, p), this means that
when
then
.
Definition 1.2 Let us have the functions
.
(1)A sequence of functions , all having the
same domain X and codomain Y is said
to converge pointwise at point xX to a given
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function f (often written as
pw
n
ff
) if and only if,
for every >0, there exists natural number p(, x)
, such that for every np(,x) we have, that
.
(2) Let (X, d), (Y, p) be two metric spaces, xX and
f n, f: XY.
It is said that the sequence
is locally
uniformly strongly convergent (or shortly a-
convergence) if for every
and >0,
such that, for every nn0(,x) and y S(x,) we have
( ( ), ( ))
n
p f y f x
, [3].
Proposition 1.3 Let f and f n be the functions that
map metric spaces (X, d) to (Y, p). If the sequence
is -convergent to the function f then
this sequence is also a- convergent to this function.
Proof. Let the sequence
be -
convergent to f. That means that if x n x on X,
then f n (x n)f(x) on Y. Since the sequence x n x
then for every there exists an n0(x,) that x n
S(x,). By the second convergence f n (x n)f(x)
we have that for nn0(x,) that
( ( ), ( )) .
nn
p f x f x
That means that the
sequence
is a- convergent to the point
xX if we substitute the x n with y in the definition
of a-convergence.
It shows that a –convergence is wider than -
convergence.
Example 1.4 We denote by K, the Cantor
continuum in [0,1], and from the construction of the
Cantor function, we know that its values are of the
form
21
2
k
nn
k
f
for k, n, and kn. We denote
( ) inf{ ( ): [0,1]\ }
k
n
f x f t t K
. Let yK and
yS(x,) we have that |
( ) ( )
k
n
f y f x
|0, from
which it follows that the
k
n
f
is a sequent to f.
Proposition 1.5 Let be (X, d), (Y, p) two metric
spaces and the function f: XY. If the sequence of
functions f n: XY for every xX, is a –
convergent than
(a) The sequence of the functions (fn(x))n is
exhaustive
(b) f(x) is continuous in every x X.
Proof. (a) By the definition of a-convergence we
have that for every >0 and /2>0 for every xX,
there exists the natural number n0(,x) and >0
such that for n n0(,x) and yS(x,), holds that
(1) ( ( ), ( )) 2
n
p f y f x
We can write that
( ( ), ( )) ( ( ), ( ))
( ( ), ( )) 22
n n n n
n
p f y f x p f y f x
p f x f x
F n are exhaustive for n n0(,x) and yS(x,).
(b) In the inequation (1), if we take instead of
>o, /3>0 for every xX there exist n1(,x) and
1>0 such that for every n> n1(,x) and yS(x,)
we have that
( ( ), ( )) 3
n
p f y f x
Also, there exist
and 2>0, such that
for every
2( , )n n x
we have
( ( ), ( )) 3
n
p f y f x
and
( ( ), ( )) .
3
n
p f x f x
The second inequation shows that from a-
convergence derives the pointwise convergence.
This allows us to write that from every /3>0, there
exists n0(,y), such that for every nn0(,y) we have
that
( ( ), ( )) 3
n
p f y f y
.
Thus, for every xX and yS(x,), where
=min{1,2} if n*=max{n0,n1,n3}, for every nn*
we take
3
3 3 3
( ( ), ( )) ( ( ), ( ))
( ( ), ( )) ( ( ), ( )) .
n
n n n
p f x f y p f x f x
p f x f y p f y f y
Definition 1.6 The net on metric(topologic) space
X is called a function that maps an ordered set of D
to X. The net is denoted (x:D) where x is a
point of the space X defined by an element of D.
In the case when D is a denumerable set, we use the
symbol{xi:iI} or (x)D. The ordered relation of D
is denoted for two elements 1,2 from D that
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12 when is 1 before 2. The point x from X is
called the limit of the net (x)D if and only if for
every neighborhood U of the point x there exists an
0D such that xU for every 0. Then we can
say that the net (x)D converges to the point x.
Unlike the sequences of elements in metric space,
the net can converge to many points at the same
time.
2 Main Results
Definition 2.1, [1], Let X be a topological space,
be a metric space, a function and a
net of functions from X to Y. We say that the
net converges to f, if for all and
all nets in x such that the net
defined by
converges to ,i.e., for all
with and , we have that
. As before we shall write
in the case where converges to f.
Definition 2.2 Let X be a topological space, be a
metric space, a function and a net
of functions from X to Y. We say that the net
converges to f if for all and every
and , we have that for , the
net converges to , i.e., for all
there exists , that for all with , we
have that .
Proposition 2.3, [1], Let X and Y be metric
spaces and also let functions . The
net converges to f (in the sense of the
previous definition) if and only if
converges to f as a sequence (i.e., in the sense of
definition 2.1).
Proposition 2.5 Let X and Y be two metric spaces
and also . The net is -
convergent to f if and only if
is -
convergent to f as a sequence according to definition
1.1.
Proof: It is evident that the -convergence of the
nets derives as a special case of the -
convergence of the sequence
Let us suppose that the sequence
a-
converges to f and y be an element such that
we want to prove that the nets -
converges to
Let’s consider the opposite assertion. The sequence
doesn’t converge to This means that
there exists an 0>0 and xX, such that for every
0<<1 and n1> n0, there exists y S (x, ) for which
10
( ( ), ( ))
n
p f y f x
. By repeating the same
procedure <
1
2
and n2n1 will find an element y
such that
to derive
, and same for
we will find y such that
to derive , and proceed
in this way indefinitely.
By an appropriate recount, we mark m=n k when k
and m0=sup {n k} then for, m >m0 there exists a
such that for it follows
that contradicts our assumption
of a-convergence of the sequence
.
Theorem 2.6 Let and the net of
functions such that map X to Y. The
following propositions are equivalent:
1)
;
2)
and the net is exhaustive.
Corollary 2.7 If the net is exhaustive and f
is -limit then the function f is a continuous
function.
Proof
The only issue that is not the same in this proof as
compared to that in Theorem 1, 3.3, [3], is the
implication .
We suppose that
meanwhile, the net of
functions fi is not exhaustive to any point .
Then, for every
we will have that for all the
open neighborhood V of the point and all
elements , will find an element such
that for every , such that for , we have:
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We mark with
the family of all the open
neighborhoods of point For two open
neighborhoods
we define if
and only if
.Thus the set
is ordered. We
denote by
, where there
exists such that The
set
is partially ordered. Let’s prove that the
set M is ordered with respect to the relation.
Actually, for
, there exists
such that and
By the axiom of choice, we can find a net
in X such that
and for all
It is obvious that for
and from the
hypothesis of -convergence of the net
This means that for , there exists V and such
that and
then
.
This contradicts the assumption and we proved that
the net is an exhaustive family.
It seems that the two concepts of -convergence and
a-convergence are equivalent. Actually, the
concepts of -convergence used in [4], in arrays in
topology, it has been shown that two convergences
x n
1
x and f m (x n)
2
f(x) are very different.
Theorem 2.8 Let X be locally compact and
a metric space and the functions , ,
where I is an ordered set. The following
propositions are equivalent:
(1) The net is -convergent to the function
f.
(2) Function f is continuous and for every compact
set the net of functions converges
uniformly to f in K.
From this Theorem derives that when X is a locally
compact space then -convergence in C
which is provided from the topology of the uniform
convergence in the compact sets. In the proof of the
theorem, we have to use regular spaces. If set I is
ordered and JI is cofinal in I then J is also ordered.
This means that when is a net then for every
J we are in conditions of the subnet .
Proof.
From the corollary (2.3), the function f is
continuous. Let K be a compact subset of X. If the
net will not converge uniformly to the
function f in K, then for every and for every
, there exist an element , such that
and , to have that .
We denote that
The set J is cofinal in I. According to the Axiom of
Choice, there exist the net in K such that
for every
Since K is a compact space, we find a subnet
and such that
As we mentioned above,
, is also applied even for subnets, so
we have
which implies that
. Regarding the continuity of the function
f, it follows the result
This
means that
which
contradicts .
We consider the net in X which
converges to the point Since X is a locally
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compact space, there exists a neighborhood of a
point , which we mark with , for which its
closure
is a compact space. We choose a
, such that for all that , we have
We define and K=
L
.
Then the space K is a closed subset of
, which is
compact. It follows from the hypothesis that the net
converges uniformly in K. As we know f is a
continuous function, so we have
for and that
proves that
.
Example 2.8.1
Let's show a case that -convergence is different
from -convergence.
We get a sequence (x n) that tends to zero in a
special way
( 1)( 2)
2
i k i k
ni
which is formed if we
go along the diagonals in the infinite table below.
For example, the third diagonal is
a31, a22, a13 the sum of the indices is 4. Thus, we
have constructed an order 1 with the pair of indices
in this way:
( , ) ( ', ')i k i k
if
''i k i k
.
If as a sequence (x n ), in X we take
1
n
xik
for
the elements in a diagonal of the table below, for
example, i + k=4, we will have that
. These terms x n can have the value 1/4 for
n=4,5,6.
Thus, the sequence is non-
decreasing monotone sequence
for It is easy to prove that
n i k
.
In the case when n, then x n0. Let’s mark
()
n n n
f x x
that goes to zero in order 1.
Let's go back to the infinite table:
11 12 13
21 22 23
31 32 33
...
...
...
... ... ... ...
a a a
a a a
a a a
Now we can choose another order 2. The sequence
sorting (x n), we do by taking the indices n
according to the "determinant rule", which allows
-convergence.
For example, we have a part of a net
x11 x21 x22 x12 … corresponding to
ranking 2, the values of the net are
.
It is clear that in this case too
n i k
but
if we fix
, we find one number,
e.g.,=100=k+i=50+50, which corresponds to e.g.,
. We see that for such n>100
11
| ( ) 0| 50 50 99
n n n
f x x
but the smallest
term
and
because
(i, k) (i, k-1) = 50+49 < n0 which defines the
definition of -convergence.
3 Conclusion
We can substitute -convergence with -
convergence even in the following statement, by
specifying some relations attributed to -
convergence and -convergence and how it is
treated to the proof of [1].
References:
[1] Gregoriades V, Papanastassiou N, The notion of
exhaustiveness and Ascoli-type theorems,
Topology and its applications, 155 (2008), pp.
1111-1128.
[2] Stoilov S., Continuous convergence, Rev.
Math. Pures Appl. 4 (1959), pp 341–344.
[3] Doda, D., Tato, A., Some local uniform,
convergences and their applications on Integral
theory, International Journal of Mathematical
Analysis, Vol. 12, 2018, no. 12, pp 631 – 645.
[4] Kelley, J., General topology, Springer- Verlag,
1975.
[5] Arens R.F., A topology for spaces of
transformations, Ann. of Math. (2) 47 (3)(1946),
pp 480–495.
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