# funcoid

## Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

**Conjecture**Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .

The main purpose here is to find a *direct* proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

**Conjecture**Let be a -separable compact reflexive symmetric funcoid and be a reloid such that

- \item ; \item .

Then .

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

## Distributivity of a lattice of funcoids is not provable without axiom of choice ★

Author(s): Porton

**Conjecture**Distributivity of the lattice of funcoids (for arbitrary sets and ) is not provable in ZF (without axiom of choice).

A similar conjecture:

**Conjecture**for arbitrary filters and on a powerset cannot be proved in ZF (without axiom of choice).

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

## Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

**Conjecture**for any composable funcoids and .

Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid

## Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

Keywords: funcoid; inward reloid; outward reloid; reloid

## Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

**Conjecture**if is a set of reloids from a set to a set .

Keywords: funcoid; infinite distributivity; reloid

## Inward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

**Conjecture**for any convex reloid .

Keywords: convex reloid; funcoid; functor; inward reloid; reloid

## Outward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

**Conjecture**for any convex reloid .

Keywords: convex reloid; funcoid; functor; outward reloid; reloid

## Distributivity of outward reloid over composition of funcoids ★★

Author(s): Porton

**Conjecture**for any composable funcoids and .

Keywords: distributive; distributivity; funcoid; functor; outward reloid; reloid

## Funcoid corresponding to inward reloid ★★

Author(s): Porton

**Conjecture**for any funcoid .

Keywords: funcoid; inward reloid; reloid

## Intersection of complete funcoids ★★

Author(s): Porton

**Conjecture**If , are complete funcoids (generalized closures) then is a complete funcoid (generalized closure).

Keywords: complete funcoid; funcoid; generalized closure