I had this theorem in mind for a long time, but formulated it exactly and proved only yesterday. Theorem $latex f \in \mathrm{C} (\mu \circ \mu^{- 1} ; \nu \circ \nu^{- 1}) \Leftrightarrow f \in \mathrm{C} (\mu; \nu)$ for complete endofuncoids $latex…
read moreI added a new easy to prove proposition to my book: Proposition An endofuncoid $latex f$ is $latex T_{1}$-separable iff $latex \mathrm{Cor}\langle f\rangle^{\ast}\{x\}\sqsubseteq\{x\}$ for every $latex x\in\mathrm{Ob}\, f$.
read moreConsider funcoid $latex \mathrm{id}^{\mathsf{FCD}}_{\Omega}$ (restricted identity funcoids on Frechet filter on some infinite set). Naturally $latex 1\in\mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega}$ (where $latex 1$ is the identity morphism). But it also holds $latex \top^{\mathsf{FCD}}\setminus 1\in\mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega}$ (where $latex 1$ is the identity morphism). This result…
read moreLess than a hour ago I discovered a new mapping from funcoids to reloids: Definition $latex (\mathsf{RLD})_X f = \bigsqcap \left\{ g \in \mathsf{RLD} \mid (\mathsf{FCD}) g \sqsupseteq f \right\}$ for every funcoid $latex f$. Now I am going to work on…
read moreFirst 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…
read moreI 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…
read moreI 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.
read moreWhat are necessary and sufficient conditions for $latex \mathrm{up}\, f$ to be a filter for a funcoid $latex f$?
read moreI’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 =…
read moreWhile 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…
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