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reloid
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 $](/files/tex/128f5af4e5b7cfd743bb0eb4fe454e040407e28f.png)
Then .
Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity
Decomposition of completions of reloids ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
- \item
![$ \operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f $](/files/tex/0844704618d467ae0507a89bdb4b215b28d57759.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g $](/files/tex/91a46d5da19358329c8cdb7351eae9cdb03c2764.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f) $](/files/tex/92e3ac66e1ae6505f78e9f443665d1bcb234fe13.png)
![$ \operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ f) $](/files/tex/528dc0c7455fad4558f8b970c989e96990663021.png)
![$ \operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g \circ f) $](/files/tex/61a4422f23782d6433a39d5ed6a48dd554f3d16f.png)
Keywords: co-completion; completion; reloid
Distributivity of inward reloid over composition of funcoids ★★
Author(s): Porton
Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid
Atomic reloids are monovalued ★★
Author(s): Porton
Keywords: atomic reloid; monovalued reloid; reloid
Composition of atomic reloids ★★
Author(s): Porton
Keywords: atomic 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
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