Normality of a quasi-uniform space on a topology is determined by the proximity induced by the quasi-uniform space
First a prelude: Taras Banakh, Alex Ravsky “Each regular paratopological group is completely regular” solved a 60 year old open problem. Taras Banakh introduces what he call normal uniformities (don’t confuse with normal topologies). My new result, proved with advanced funcoids theory (and never tried to prove it with basic general topology): Whether a uniformity on […]
Virtual mathematics conferences opened!
I have just created a new wiki Web site, which is a virtual math conference, just like a real math meeting but running all the time (not say once per two years). https://conference.portonvictor.org Please spread the word that we have a new kind of math conference. Please post references to your articles, videos, slides, etc.
New proof of Urysohn’s lemma
I present a new proof of Urysohn’s lemma. Well, not quite: my proof is dependent on an unproved conjecture. Currently my proof is present in this PDF file. The proof uses theory of funcoids.
A math question
What are necessary and sufficient conditions for $latex \mathrm{up}\, f$ to be a filter for a funcoid $latex f$?
New easy theorem and its consequence
I’ve added to my book a new easy to prove theorem and its corollary: Theorem If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f$ is a filter. Corollary If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f = \mathrm{up} (\mathsf{RLD})_{\mathrm{out}} f$. If $latex f$ is a (co-)complete reloid then $latex \mathrm{up}\, […]
A new conjecture
While writing my book I forgot to settle the following conjecture: Conjecture $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex f$ and $latex g$. Note that the similar statement about reloids is quite obvious.
A new (but easy to prove) theorem in my research book: Theorem Let $latex \mu$ and $latex \nu$ be endomorphisms of some partially ordered dagger precategory and $latex f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu)$ be a monovalued, entirely defined morphism. Then $latex f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).$
$T_4$-funcoids
I have added to my free ebook a definition of $latex T_4$-funcoids (generalizing $latex T_4$ topologies). A funcoid $latex f$ is $latex T_4$ iff $latex f \circ f^{- 1} \circ f \circ f^{- 1} \sqsubseteq f \circ f^{- 1}$. This can also be generalized for pointfree funcoids.
Reexamined: Normal quasi-uniformity elegantly defined
Earlier I’ve conceived an algebraic formula to characterize whether a quasi-uniform space is normal (where normality is defined in Taras Banakh sense, not in the sense of underlying topology being normal). That my formula was erroneous. Today, I have proved another formula for this (hopefully now correct): Theorem An endoreloid $latex f$ is normal iff […]
New easy theorem and a couple of conjectures
I added to my free ebook a new (easy to prove) theorem and a couple new conjectures: Theorem For every reloid $latex f$: $latex \mathsf{Compl} (\mathsf{FCD}) f = (\mathsf{FCD}) \mathsf{Compl} f$; $latex \mathsf{CoCompl} (\mathsf{FCD}) f = (\mathsf{FCD}) \mathsf{CoCompl} f$. Conjecture $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{in}} g=(\mathsf{RLD})_{\mathsf{in}}\mathsf{Compl} g$; $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{out}} g=(\mathsf{RLD})_{\mathsf{out}}\mathsf{Compl} g$;.