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compact space
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
.
![$ 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 $](/files/tex/630751f9d9f5276b67a64fc57d858c975ce7f9e4.png)
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
Conjecture Let
be a
-separable compact reflexive symmetric funcoid and
be a reloid such that
![$ 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
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