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18.5. GENERALIZED LIMIT

278

18.4. Limit

Definition

1464

.

lim

µ

f

=

a

iff

f

µ

→↑

Src

µ

{

a

}

for a

T

2

-separable funcoid

µ

and a non-empty funcoid

f

such that Dst

f

= Dst

µ

.

It is defined correctly, that is

f

has no more than one limit.

Proof.

Let lim

µ

f

=

a

and lim

µ

f

=

b

. Then im

f

v h

µ

i

@

{

a

}

and im

f

v

h

µ

i

@

{

b

}

.

Because

f

6

=

FCD

(Src

f,

Dst

f

)

we have im

f

6

=

F

(Dst

f

)

;

h

µ

i

@

{

a

} u h

µ

i

@

{

b

} 6

=

F

(Dst

f

)

;

Src

µ

{

b

} u

µ

1

h

µ

i

@

{

a

} 6

=

F

(Src

µ

)

;

Src

µ

{

b

} u

µ

1

µ

@

{

a

} 6

=

F

(Src

µ

)

; @

{

a

}

µ

1

µ

@

{

b

}

. Because

µ

is

T

2

-separable we have

a

=

b

.

Definition

1465

.

lim

µ

B

f

= lim

µ

(

f

|

B

).

Remark

1466

.

We can also in an obvious way define limit of a reloid.

18.5. Generalized limit

18.5.1. Definition.

Let

µ

and

ν

be endofuncoids. Let

G

be a transitive per-

mutation group on Ob

µ

.

For an element

r

G

we will denote

r

=

FCD

(Ob

µ,

Ob

µ

)

r

.

We require that

µ

and every

r

G

commute, that is

µ

◦ ↑

r

=

r

µ.

We require for every

y

Ob

ν

ν

w h

ν

i

@

{

y

} ×

FCD

h

ν

i

@

{

y

}

.

(19)

Proposition

1467

.

Formula (

19

follows from

ν

w

ν

ν

1

.

Proof.

Let

ν

w

ν

ν

1

. Then

h

ν

i

@

{

y

} ×

FCD

h

ν

i

@

{

y

}

=

h

ν

i

@

{

y

} ×

FCD

h

ν

i

@

{

y

}

=

ν

(

Ob

ν

{

y

FCD

Ob

ν

{

y

}

)

ν

1

=

ν

◦ ↑

FCD

(Ob

ν,

Ob

ν

)

(

{

y

} × {

y

}

)

ν

1

v

ν

1

FCD

Ob

ν

ν

1

=

ν

ν

1

v

ν.

Remark

1468

.

The formula (

19

usually works if

ν

is a proximity. It does not

work if

µ

is a pretopology or preclosure.

We are going to consider (generalized) limits of arbitrary functions acting from

Ob

µ

to Ob

ν

. (The functions in consideration are not required to be continuous.)

Remark

1469

.

Most typically

G

is the group of translations of some topological

vector space.

Generalized limit

is defined by the following formula:

Definition

1470

.

xlim

f

def

=

n

ν

f

◦↑

r

r

G

o

for any funcoid

f

.

Remark

1471

.

Generalized limit technically is a set of funcoids.

We will assume that dom

f

w h

µ

i

@

{

x

}

.

Definition

1472

.

xlim

x

f

= xlim

f

|

h

µ

i

@

{

x

}

.