📚 Recommended Mathematics Books
Topology (Munkres) | General Topology (Engelking) | Counterexamples in Topology | Rudin's AnalysisAs an Amazon associate, I earn from qualifying purchases.
I’ve found a counterexample to the following conjecture:
Statement For every composable funcoids $latex f$ and $latex g$ we have
$latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$
The counterexample is $latex f=a\times^{\mathsf{FCD}} \{p\}$ and $latex g=\{p\}\times^{\mathsf{FCD}}a$, $latex H=1$ where $latex a$ is an arbitrary nontrivial ultrafilter and $latex p$ 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.
🔬 Advanced Mathematics References
- Sheaves in Geometry and Logic
- Categories for the Working Mathematician
- Stone Spaces
- Algebraic Topology (Hatcher)
- Concrete Mathematics
As an Amazon associate I refer fromm qualifying purchases.
Can you be more specific about the content of your article? After reading it, I still have some doubts. Hope you can help me.