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February 1, 2014 / porton

An attempt to generalize a theorem failed

Using “compactness of funcoids” which I defined earlier, I’ve attempted to generalize the classic general topology theorem that compact topological spaces and uniform spaces bijectively correspond to each other.

I’ve resulted with the theorem

Theorem Let f is a T_1-separable (the same as T_2 for symmetric transitive) compact funcoid and g is an reflexive, symmetric, and transitive endoreloid such that ( \mathsf{FCD}) g = f. Then g = \langle f \times f \rangle \uparrow^{\mathsf{RLD}} \Delta.

But wait, reflexive, symmetric, and transitive endoreloid is practically the same as a uniform space.

So my theorem is about uniform spaces, just like as the classic theorem. I haven’t succeeded to generalize, I’ve just formulated and proved the same classical well known theorem.

A sad for me conclusion: My theory has not added value for the case of compact spaces. In this case my theory just coincides with classic general topology.

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