![](/files/happy5.png)
funcoid
Several ways to apply a (multivalued) multiargument function to a family of filters ★★★
Author(s): Porton
![$ \mathcal{X} $](/files/tex/dcd7ae9cf0009f744cb554a2a549667e2a95aed0.png)
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
3. .
Keywords: funcoid; function; multifuncoid; staroid
Direct proof of a theorem about compact funcoids ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ T_1 $](/files/tex/d9a989037d7243d9036b9c8165c72e0331991a8c.png)
![$ T_2 $](/files/tex/55e29109946a61a46e41f972a62209d3dbd4e96c.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ ( \mathsf{\tmop{FCD}}) g = f $](/files/tex/f68984666c5a1553c57c071dd482b9e3f1869eb4.png)
![$ g = \langle f \times f \rangle^{\ast} \Delta $](/files/tex/630751f9d9f5276b67a64fc57d858c975ce7f9e4.png)
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ T_1 $](/files/tex/d9a989037d7243d9036b9c8165c72e0331991a8c.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
- \item
![$ ( \mathsf{\tmop{FCD}}) g = f $](/files/tex/f68984666c5a1553c57c071dd482b9e3f1869eb4.png)
![$ g \circ g^{- 1} \sqsubseteq g $](/files/tex/128f5af4e5b7cfd743bb0eb4fe454e040407e28f.png)
Then .
Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity
Distributivity of a lattice of funcoids is not provable without axiom of choice ★
Author(s): Porton
![$ \mathsf{FCD}(A;B) $](/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
A similar conjecture:
![$ a\setminus^{\ast} b = a\#b $](/files/tex/c6fc3b6da0655ddeaeffe670703a33edcb4650f6.png)
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
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
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
Keywords: funcoid; infinite distributivity; reloid
Inward reloid corresponding to a funcoid corresponding to convex reloid ★★
Author(s): Porton
Keywords: convex reloid; funcoid; functor; inward reloid; reloid
Outward reloid corresponding to a funcoid corresponding to convex reloid ★★
Author(s): Porton
Keywords: convex reloid; funcoid; functor; outward reloid; reloid
Distributivity of outward reloid over composition of funcoids ★★
Author(s): Porton
Keywords: distributive; distributivity; funcoid; functor; outward reloid; reloid
Funcoid corresponding to inward reloid ★★
Author(s): Porton
Keywords: funcoid; inward reloid; reloid
![Syndicate content Syndicate content](/misc/feed.png)