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8.8. WHAT UNIFORM SPACES ARE

196

Proposition

1058

.

For composable reloids

f

and

g

it holds

1

. Compl(

g

f

)

w

(Compl

g

)

(Compl

f

)

2

. CoCompl(

g

f

)

w

(CoCompl

g

)

(CoCompl

f

).

Proof.

1

. (Compl

g

)

(Compl

f

)

v

Compl((Compl

g

)

(Compl

f

))

v

Compl(

g

f

).

2

. By duality.

Conjecture

1059

.

For composable reloids

f

and

g

it holds

1

. Compl(

g

f

) = (Compl

g

)

f

if

f

is a co-complete reloid;

2

. CoCompl(

f

g

) =

f

CoCompl

g

if

f

is a complete reloid;

3

. CoCompl((Compl

g

)

f

) = Compl(

g

(CoCompl

f

)) = (Compl

g

)

(CoCompl

f

);

4

. Compl(

g

(Compl

f

)) = Compl(

g

f

);

5

. CoCompl((CoCompl

g

)

f

) = CoCompl(

g

f

).

8.8. What uniform spaces are

Proposition

1060

.

Uniform spaces are exactly reflexive, symmetric, transitive

endoreloids.

Proof.

Easy to prove using theorem

1003

.