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11.5. CONTINUITY OF A RESTRICTED MORPHISM

223

2

3

Let

B

be a neighborhood of

f

(

x

). Then there is an open neighborhood

B

0

B

of

f

(

x

).

f

1

B

0

is open and thus is a neighborhood of

x

(

x

f

1

B

0

because

f

(

x

)

B

0

). Consequently

f

1

B

is a neighborhood

of

x

.

Alternative proof of

2

4

:

http://math.stackexchange.com/a/1855782/4876

11.4.

C(

µ

µ

1

, ν

ν

1

)

Proposition

1181

.

f

C(

µ, ν

)

f

C

00

(

µ

µ

1

, ν

ν

1

) for endofuncoids

µ

,

ν

and monovalued funcoid

f

FCD

(Ob

µ,

Ob

ν

).

Proof.

Let

f

C(

µ, ν

).

X

[

f

µ

µ

1

f

1

]

Z

p

atoms

F

:

X

[

µ

1

f

1

]

p

p

[

f

µ

]

Z

⇔ ∃

p

atoms

F

: (

p

[

f

µ

]

X

p

[

f

µ

]

Z

)

p

atoms

F

:

(

p

[

ν

f

]

X

p

[

ν

f

]

Z

)

p

atoms

F

:

h

f

i

p

[

ν

]

X

∧ h

f

i

p

[

ν

]

Z

X

[

ν

ν

1

]

Z

(taken into account monovalued-

ness of

f

and thus that

h

f

i

p

is atomic or least). Thus

f

µ

µ

1

f

1

v

ν

ν

1

that is

f

C

00

(

µ

µ

1

, ν

ν

1

).

Proposition

1182

.

f

C

00

(

µ

µ

1

, ν

ν

1

)

f

C

00

(

µ, ν

) for complete

endofuncoids

µ

,

ν

and principal funcoid

f

FCD

(Ob

µ,

Ob

ν

), provided that

µ

is

reflexive, and

ν

is

T

1

-separable.

Proof.

f

C

00

(

µ

µ

1

, ν

ν

1

)

f

µ

µ

1

f

1

v

ν

ν

1

(reflexivity of

µ

)

f

µ

f

1

v

ν

ν

1

f

µ

1

f

1

v

ν

ν

1

h

f

µ

1

f

1

i

X

v h

ν

i

h

ν

1

i

X

Cor

f

µ

1

f

1

X

v

Cor

h

ν

i

h

ν

1

i

X

h

f

µ

1

f

1

i

X

v

Cor

h

ν

i

h

ν

1

i

X

(

T

1

-separability)

⇒ h

f

µ

1

f

1

i

X

v

h

ν

1

i

X

for any typed set

X

on Ob

ν

. Thus

f

C

00

(

µ

µ

1

, ν

ν

1

)

f

µ

1

f

1

v

ν

1

f

µ

f

1

v

ν

f

C

00

(

µ, ν

).

Theorem

1183

.

f

C(

µ

µ

1

, ν

ν

1

)

f

C(

µ, ν

) for complete

endofuncoids

µ

,

ν

and principal monovalued and entirely defined funcoid

f

FCD

(Ob

µ,

Ob

ν

), provided that

µ

is reflexive, and

ν

is

T

1

-separable.

Proof.

Two above propositions and theorem

1175

.

11.5. Continuity of a restricted morphism

Consider some partially ordered semigroup. (For example it can be the semi-

group of funcoids or semigroup of reloids on some set regarding the composition.)

Consider also some lattice (

lattice of objects

). (For example take the lattice of set

theoretic filters.)

We will map every object

A

to so called

restricted identity

element

I

A

of the

semigroup (for example restricted identity funcoid or restricted identity reloid). For

identity elements we will require

1

.

I

A

I

B

=

I

A

u

B

;

2

.

f

I

A

v

f

;

I

A

f

v

f

.

In the case when our semigroup is “dagger” (that is is a dagger precategory) we

will require also (

I

A

)

=

I

A

.

We can define restricting an element

f

of our semigroup to an object

A

by the

formula

f

|

A

=

f

I

A

.

We can define

rectangular restricting

an element

f

of our semigroup to objects

A

and

B

as

I

B

f

I

A

. Optionally we can define direct product

A

×

B

of two

objects by the formula (true for funcoids and for reloids):

f

u

(

A

×

B

) =

I

B

f

I

A

.