Theorem 34. A symmetric transitive reloid is totally bounded iff its Cauchy space is totally
bounded.
Proof.
⇒. Let F be a proper filter on Ob ν and let a ∈ atoms F. It’s enough to prove that a is Cauchy.
Let D ∈ GR ν. Let also E ∈ GR ν is symmetric and E ◦ E ⊆ D. There existsa finite subset
F ⊆ Ob ν such that hE iF = Ob ν. Then obvio usly exists x ∈ F such that a ⊑ ↑
Ob ν
hE i{x},
but hE i{x} × hE i{x} = E
−1
◦ ({x} × {x}) ◦ E ⊆ D, thus a ×
RLD
a ⊑ ↑
RLD(Ob ν;Ob ν)
D.
Because D was taken arbit rary, we have a ×
RLD
a ⊑ ν that is a is Cauchy.
⇐. Suppose that Cauchy space associated with a reloid ν is totally bounded but the reloid
ν isn’t totally bounded. So the re exists a D ∈ GR ν such that (Ob ν) \ hDiF
∅ for every
finite set F .
Consider the filter base
S = {(Ob ν) \ hD iF | F ∈ P Ob ν , F is finite}
and the filter F =
d
h↑
Ob ν
iS generated by this base. The filter F is proper be cause
intersection P ∩ Q ∈ S for every P , Q ∈ S and ∅
S. Thus there exists a Cauchy (for our
Cauchy space) filter X ⊑ F that is X ×
RLD
X ⊑ ν.
Thus there exists M ∈ X such that M × M ⊆ D . Let F be a finite subset of Ob ν.
Then (Ob ν) \ hDiF ∈ F ⊒ X . Thus M
(Ob ν) \ hDiF and so there exists a point
x ∈ M ∩ ((Ob ν) \ hDiF ).
hM × M i{p} ⊆ hDi{x} for every p ∈ M; thus M ⊆ hDi{x}.
So M ⊆ hD i(F ∪ {x}). But this means that M ∈ X does not intersect (Ob ν) \
hDi(F ∪ {x}) ∈ F ⊒ X , what is a contradiction (taken into account that X is proper).
http://math.stackexchange.com/questions/104696/pre-compactness-total-boundedness-and-
cauchy-sequential-compactness
10 Totally bounded funcoids
Definition 35. A funcoid ν is totally bounde d iff
∀X ∈ Ob ν ∃X ∈ F
Ob ν
: (0
X ⊑ ↑
Ob ν
X ∧ X ×
FCD
X ⊑ ν).
This can be rewritten in elementary terms (without using fu nc oidal product:
X ×
FCD
X ⊑ ν ⇔ ∀P ∈ ∂X : X ⊑ hν iP ⇔ ∀P ∈ ∂X , Q ∈ ∂X : P [ν]
∗
Q ⇔ ∀P , Q ∈ Ob ν:
(∀E ∈ X : (E ∩ P
∅ ∧ E ∩ Q
∅) ⇒ P [ν]
∗
Q).
Note that probably I am the first p e rson which has written the above formula (for p roximity
spaces for instance) explicitly.
11 On principal low fi lter spaces
Definition 36. A low filter sp ace (U ; C ) is principal when all filters in C are principal.
Definition 37. A low filter sp ace (U ; C ) is reflexive when ∀x ∈ U : ↑
U
{x} ∈ C .
Proposition 38. Having fixed a set U , principal reflexive low filter spaces on U bijectively cor-
respond to principal reflexive symmertic endoreloids on U.
Proof. ??
http://math.stackexchange.com/questions/701684/union-of-cartesian-squares
On principal low filter spaces 5