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October 15, 2017 / porton

A counterexample to my conjecture

I’ve found a counterexample to the following conjecture:

Statement For every composable funcoids f and g we have

H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .

The counterexample is f=a\times^{\mathsf{FCD}} \{p\} and g=\{p\}\times^{\mathsf{FCD}}a, H=1 where a is an arbitrary nontrivial ultrafilter and p is an arbitrary point.

I leave the proof that it is a counterexample as an easy exercise for the reader (however I am going to add the proof to my book soon).

That the conjecture failed invalidates my new proof of Urysohn’s lemma which was based on this conjecture.


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