background image

1. ON “EACH REGULAR PARATOPOLOGICAL GROUP IS COMPLETELY REGULAR” ARTICLE

67

Prove

ν

1

h

U

q

i

A

v

ν

1

ν

1

h

U

r

i

A

for any

q < r

in

Q

2

.

FiXme

:

Can be easily rewritten with the formula

h

ν

i

ν

1

h

U

q

i

A

v

ν

1

h

U

r

i

A

in-

stead. It may extend to non-complete funcoids.

There is such

l

that 0 =

q

l

< r

l

= 1 and

q

i

=

r

i

for all

i < l

.

It follows

l

q

6

=

l

l

r

.

Consider variants:

l

q

< l

.

ν

1

h

U

q

i

A

v

ν

1

D

U

l

q

. . .

U

q

1

q

lq

1

E

A

=

ν

1

D

U

r

lq

l

q

. . .

U

r

1

1

E

A

v

h

ν

1

i

U

r

l

1

l

1

. . .

U

r

1

1

A

v

h

ν

1

i

ν

1

h

U

r

l

l

U

r

l

1

l

1

. . .

U

r

1

1

i

A

=

h

ν

1

i

ν

1

h

U

r

i

A

(use

U

r

l

l

up(

FCD

)

µ

by theorem 992).

l < l

q

. Inclusions

U

k

U

k

v

U

k

1

for

l < k

l

q

+ 1 guarantee that

U

l

q

+1

U

l

q

. . .

U

l

+1

v

U

l

and then

ν

1

h

U

q

i

A

v

ν

1

D

U

q

lq

l

q

. . .

U

q

1

1

E

A

v

ν

1

ν

1

D

U

q

lq

+1

l

q

+1

U

q

lq

l

q

. . .

U

q

1

1

E

A

=

ν

1

ν

1

D

U

l

q

+1

U

q

lq

l

q

. . .

U

0

l

. . .

U

q

1

1

E

A

v

ν

1

ν

1

h

U

l

U

q

l

1

l

1

. . .

U

q

1

1

i

A

v

ν

1

h

ν

1

i

U

r

l

l

U

r

l

1

l

1

. . .

U

r

1

1

A

v

h

ν

1

i

ν

1

U

r

lr

l

r

. . .

U

r

1

1

A

=

ν

1

ν

1

h

U

r

i

A

.

Define

f

by the formula

f

(

z

) = inf

{

1

} ∪

n

q

Q

2

z

∈h

ν

1

i

h

U

q

i

A

o

.

It is clear?? that

A

v

f

1

{

0

}

and

f

1

[0; 1[

v

S

q

Q

2

ν

1

h

U

q

i

A

=

S

r

Q

2

h

ν

1

i

ν

1

h

U

r

i

A

v

ν

1

h

ν

1

i

h

U

0

i

A

.

To prove that the map

f

:

X

[0

,

1] is continuous, it suffices to check that

for every real number

a

]0; 1[ the sets

h

f

1

i

[0;

a

[ and

f

1

]

a

; 1] are open. This

follows from the equalitites

f

1

[0;

a

[=

S

Q

2

3

q<a

ν

1

ν

1

h

U

q

i

A

and

f

1

]

a

; 1] =

S

Q

2

3

r>a

(

X

\

h

ν

1

i

h

U

r

i

A

).

How the formulas for normal (

T

4

) topological spaces and normal quasi-

uniformities are related? Maybe this works: Replacing

ν

µ

µ

1

,

µ

1 makes

ν

ν

1

v

ν

1

(

FCD

)

µ

µ

µ

1

µ

µ

1

v

µ

µ

1

.

https://www.researchgate.net/project/The-lattice-LG-of-group-topologies