# On the representation of uniform structures by extended reticles

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## S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze

e

o

### 4 (1969), p. 553-561

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### OF UNIFORM STRUCTURES BY EXTENDED RETICLES

GIOVANNI VIDOSSICH

The extended reticles

were introduced

(j.

in

as a tool for

of uniform

### structures ;

in that paper it is

### proved

that some uniformities admit of a

### representation by

suitable extended re- ticles and

pp.

### 353-354])

it is left as an open

to find if every

has such a

This

has

an anal-

ogy with the

in

Res. Probl.

if every

has a basis

of

discrete

but the two

### problems

are different because the former has a

### topological premise -

the definition of extended reticle - and a uniform

### conclusion,

while the latter has an

uniform nature.

In the

existence and

theorems on the

to

a

### uniformity by

extended reticles are

means

of the notion of

classes

introduced in

Main

result:

### Every compatible uniformity

on a unifonnizable

space X is the

### uniformity

associated to an extended reticle iff X is

which

a

of G.

a result

### quite

natural wlen one thinks to the

of the

### problem.

The uniformities associated to the universal extended reticle

its

### definition,

as well as for those of other

### uncustoluary

terms used in this re-

of a

space X have

as

and

### [5]

show. But from the results of this paper and from our

### knowledge,

it seems that we cannot say the same

### tliing

The best matter to do it seems to be to define the uniformities Uk

and

from

finite or

finite

### %.reducible (= 1R (X)~reducible)

open covers and to make no use of the

Pervenuto alla Redazione il 21 Novenbre 1967.

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554

notion of extended reticle: this leads to many

in the

when 0.4.

is

in the

of the very

### complicate original

de-

finition of « to be

o.

of

### known

results.

In this section some definitions and results from

and

are

from

### [1; 2],

in order to make the paper self contained

as much as

We shall use

- without any

- the

### ter,ntinotogy

and the notations

this section.

means a set of

### entourages

and not of uniform covers.

Let

### (~~ 11)

be a uniform space. For every

B C

B is said to be

a

of

written

iit’ there is

### W E n

such that A cover

of X is said

### to be

u-reducible iff there is a cover of X such that

for all

### i E I ;

then every cover with the

of

is said a

of

E r·

Two uniformities

I

### ti2

on X are said to be p

iff’

for all In

### [1]

it is shown that is

to is an

### equivalence

relation on the set of all uniformities

with a

### given topology

on a set X and that every its

called

### p-equ’ivalence class,

contains a coarsest

### uniformity

which is also the

### unique precompact uniformity

in the class. This

### uniformity

has as basis all sets of the form

where is a finite

open cover of

U

### being

any member of the

class.

Let X be a

and

### O (X)

the sets of all closed and

### respectively

open substes of X. An extended reticle on X is a subset

such that:

(4)

c I is in

and is

finite ==-&#x3E;

### ( U Ai, U B,)E1R.

1,£1 i E I

For every infiuite cardinal

a is defined

with

### Clearly

every extended reticles is a k-exten(led reticle for all infinite

becames

to

### (ERv)k

for tlcis k. Therefore the

### following

definitions and results

### apply

also to

extended reticles

=

or li

.

Let

### 1R

be a k-extended reticle on X. If is an open cover of

### X,

then a closed cover is said to be an

### 1R

for all An open cover of X is said to be iff it has some

### 1R - red uction.

For any cover of X and every ve pose

0.1

Lemma

Let

### 1R

be a k-extended reticle on a

space

k any

Then

every

### locally finite 1R-j’edu-

cible open cover such

and every its

there is a

open cover

X such

and St

all i E I.

0.2.

Lemma

In the

1 is an

open cover

then there is a

co-

ver

n

1

X such that

all n N.

Let

### 1R

be a k-estended reticle on X and

### ko

any infinite cardinal k.

As

Lemma

the set of all

finite

### R-reducible

open

covers of X whose

set has

~

### ko

constitutes a uniform

### covering system

in the sense of J. W.

the

unifor-

is called the

associated to

and denoted

we

the

notations

(5)

556

When

is an extended

the

associated to

written

is the

essociated to

with k

0.3

Th. 1 and

In the

we have :

WE

there is a

cover

E I

X such that

and

X

w.

For every

E I in

such

k and

I is

### locally finite,

there is W E tik such that W

C

all i E I.

If

### A,

B are two subsets of a

space

### X,

then A is said to be

### U-contained

in B iff’ there is a sequence - called

sequence -

### (( Um )n o )m-o

of finite families of open subsets of X such that :

tl= m=o

The set

E

is

in

is an extended

reticle on

### X,

called the universal extended i-eticle

tlce

### topological

space X and denoted

From 0.1.

from

Lemma 1

Lem-

ma

it follows the

0.4. For every

### topological

space X and every

E

&#x3E;C

the

statements are

A is

in B.

There is an

W

a

on X with

less

than that

which we have W

B.

### (3)

There is a continuous

X .--~

suclz

and

_

0.5.

Def.

Lemma 2

8, Cor. 1 to

For eV6t.y

space X tlae

state1nents are

l

X is

on X

less

than that

X is

open cover

X has a

subcover.

countable

open cover

X has a

subcover.

(6)

1.

### Uniqueness theorems.

1.1. LEMMA : Let k be an

X a

### topological

space and reticle on X. Then

every subsets

B

it

the existence

E

such that A c

and

PROOF: Let

be such that

0.3.

there is

a

finite

open cover of .~

and

W. Let be any

of and 7*==

### [

t c r -

For every x E X there is such that

then

X

and Z = = which

### implies

From this and it

I

follows that it suffices to choose A*‘ =

and B*‘ =

### U B; I

/

1EI e*

1.2. THEOREM : Let X be a

a

### p-equi-

valeaice class on X and k an

cardinal.

### Then P

may contain. at most

one

which is the

### k-uniformity

associated to a k-extended reticle.

PROOF : Assume that

E

### ~

are the k-uniformities associated to two

k-extended

to

and From

it

0.3.

and hence Thus

1.1 there is

E

### IR2

such and B. Then for every

finite

open cover of

there is an

open cover

E r with

for all i

which

is

finite. This

0.3.

that

is less hne than

The same reasons

### imply

the converse and so

=

### u2 I

The above theorem

that we may find

one member

which is the

### uniformity

associated to an extended reticle. This result is

the

Theorem

whose

needs the

lemma which is

a famous

in the

of the

### paracompact-

ness of metrizable spaces.

1.3. Let

be a

### uniform

space. For every WE U there is a

open

X

### re,fining

PROOF : Let d be a

on X whose

### uniformity

is less fine than U and contains W. Because is refined

a

cover

of

we may

the

of

Lemma

to ob-

tain our

### goal. 1

2. dnnali della Scuola Norm. Sup.

(7)

558

The converse of the

theorem fails

### by

virtue of 2.5.

1.4. THEOREM : Let X be a

space

a

class on X.

### If 9

contains an element U which is the

### uniformity

associated to

an extended reticle

on

then 11 is the

in

PROOF : Let

be

and

W E

such that

and

W is

### symmetric. By 1.3,

there is a a-discrete

open cover I

of X finer than

Let

I be a

a

of I .

there is

I in

such that

C

and I -

for all i E I.

0.3.

X

### Bi*‘

E u. From this and

i c I s I

X

W

X W

### [x] c Wo

it follows that tt is finer than

EI t t -

2.

The

### following proposition

shows that all uniformities defined in

### [3, §. 31 by

means of extended reticles - i. e. Uoo

and 1ik

an infinite

-

to the same

### p-equivalence

class.

2.1. PROPOSITION: Lot X be a

space a

### p-equiva-

lence class on X.

### If 9

contains an element which is the

### k-uniforntity

associated

to a k-extended reticle

on X

an

card inal

then

### 1&#x3E;

contains also the

associated to

every

cardinal k’ In

contains the

### uniforntity

associated to an extended reticle

contains the k

associated to

every

### in, fiuite

cardinal k.

PROOF : Let k’ be any infinite cardinal S k. From the definitions it follows that is less fine than Uk

### (1R)

and so it is sufficient to show that A

B.

### By 1.1, implies

the

existence such that

### By

and so The latter assertion follows from the

### No.1

2.2 THEOREM : Let X be a

space. For every

U

with the

### topology of X,

the set

is an extended reticle on X and 11 is - at tlae same time - the

every

cardinal

the

associated to

(8)

PROOF : It is

see

Th.

that

fulfils axioms

To prove

let

in

such

that

is

finite.

### Let Io

be the set of all i E I such that ø. In order to show that

is

### finite, assume Io

to be infinite and argue for a contradiction.

to each

there

a continuous map

It such that =

and

Since

there is an

sequence in

Since is

00

is a continuous

X-+ R. But

### f

is unbounded since

n= 1

= what is

the

of X. Hence

### 10

must be finite. Then we have

Thus

### (ERV) holds,

and is an extended reticle on X. Let

be the.

- 0

associated to

### ’R (tij,

From A B it follows

CCu

hence

- 0 - 0

and so,

0.3.

A cCno

which

A

### B;

vice-

versa, from A B it

the existence of

such that and B*

what

A B

definition of

Thus U and

to the same

class

which -

### by

2.1

- contains also the

associated to

### 1R (U)

for all infinite cardinal

k.

0.5 all members

are

and

must

coincide

a theorem of

recalled

contains a

### unique precompact uniformity. 1

2.3 THEOREM : The

space X is

every

on X whose

is less

than that

X is the

associated to an

### S0-extended

reticle on X. In order that X be

it is

### sufficient

and necessary that the

less

than that

is the

associated to an

reticle on X.

PROOF : The

follows from

since a

less fine

than a

is

Let

### 1R (X)

be the universal extended reticle on

the p

class contain-

Uoo

and U

the

of

Because Uoo

is the finest

on X

### having topology

less fine than that of

is the finest

on X

### having topology

less

fine than that of X. Therefore we prove both statements of the theorem at the same time

as we

that 1B

is the

associated

to an

reticle on X.

and

= u

### (~).

Let 1 be a countable

open

of X.

0.2 there is a

finite

open

(9)

560

of X such that

for all n

### =1,

... , oo. Let be an

duction of

0.3.

there is W E

= U

such that

W

~

### On

for every n. Since U

is

there is ... ,

### 9

X

k k k

such that X = i=l

W

If xi E

for all

then X c -

W

Therefore is

0.5 «

---~

### (1) ».1

2.4. COROLLARY : Let X be a

space and k an

cardinal.

Then X is

every

on X

less

than that

### of

X is the associated to a k-extended reticle

the

### uniformity

associated to an extended reticle on

### X).

In order that X be

it is

that this

the

on X

les8

than thcct

X.

PROOF : The

### necessity

follows as for 2.3.

We use the no-

tations of the

### proof

of 2.3. Because ti

is the

### k.uniformity (resp. : unifor- mity)

associated to a k-extended

reticle

on

2.1

contains

### US0 (1R).

From the definitions it follows that

### US0 (1R)

is less fine than 1t and hence

= since

### U (9))

is the coarsest unifor-

in

### SP.

Thus the conclusion follows from

The

### following proposition,

as well as 2.3 and

a

of G.

### AQUARO explained

in the introduction of

### the,

paper ; it also shows that the converse of 1.4 does not hold.

2.5. PROPOSITION : There is a

on R

a

### finest uniformity

such that no element

it is the

nor the

### k-uniformity

associated to an extended reticle or,

### respectiroely,

a k-extended reticle on

k

an

cardinal.

PROOF : Let

be the

defined

### by

the euclidean metric on

the p

class

### containing Uo . By

a resuJt of Yu - M. SHIR-

NOV cited in

p.

### 204], 7&#x3E;

contains a finest

### uniformity. By 2.1,

to prove

our assertion it suffices to show that no element

may be the

associated to an

### No.extended

reticle on R. Assume that there is an

reticle

on R such that

E

For every n

define

=

and

E

For every n E N

and so No

there is

E

such that

and

1 is

fi-

(10)

nite and

00

0.3.

that conse-

Y there is

such

=

&#x3E;

### Wa [A] g

B. Let n E N be such that

a, and let x E

Since

### lva [A ] z B C U Bn ,

there is n’ E N such that x E

### Bn’.

From this

.

and the definitions of

and

it follows :

2 and a

### 2, necessarily

n’ 2 2. Therefore a

which

7t = n’. Then

which is

if t

then n

### + t E Wa B~ Bn · ~

REFERENCES

1. E. M. ALFSEN and J. E. FENSTAD : On the equivalence between proximity structures and

totally bounded uniform structures, Math. Scand. 7 (1959), 357-360.

2. E. M. ALFSEN and J. E. FENSTAD: A note on completion and compactification, Math.

Scand. 8 (1960), 97-104.

3. G. AQUARO : Ricovrimenti aperti e strutture uniformi sopra uno spazio topologico, Ann. Mat

Pura Appl. 47 (1959), 319-390.

4. G. AQUARO: Spazii collettivamente normali ad estensione di applicazione continue, Riv. Mat.

Univ. Parma 2 (1961), 77-90.

5. G. AQUARO : Convergenza localmente quasi-uniforme ed estensione di Hewitt di uno spazio completamente regolare, Ann. Sc. Norm. Sup. Pisa 16 (1962), 207- 212.

6. N. BOURBAKI: Topologie générale, ch. 9, Hermann Paris, 1958

### (2nd

ed.).

7. J. R. ISBELL: Uniform Spaces, Amer. Math. Soc., Providence, 1964.

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