A funcoid related to directed topological spaces
The following problem arose from my attempt to re-express directed topological spaces in terms of funcoids. Conjecture Let $latex R$ be the complete funcoid corresponding to the usual topology on extended real line $latex [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\}$. Let $latex \geq$ be the order on this set. Then $latex R\sqcap^{\mathsf{FCD}}\mathord{\geq}$ is a complete funcoid.
Two equivalent conjectures
I have added to my book a short proof that the following two conjectures are equivalent: Conjecture $latex \mathrm{Compl}\,f \sqcap \mathrm{Compl}\,g = \mathrm{Compl}(f\sqcap g)$ for every reloids $latex f$ and $latex g$. Conjecture Meet of every two complete reloids is complete.
A new conjecture
While writing my book I overlooked to consider the following statement: Conjecture $latex f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $latex f$ and a set $latex S$ of funcoids of appropriate sources and destinations.
A new theorem about generalized continuity
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 $latex \mu$, $latex \nu$ and principal monovalued and entirely defined funcoid $latex f \in […]
A new easy theorem
I 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 $latex x\in\mathrm{Ob}\, f$.
A vaguely formulated problem
Consider 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 is not hard to prove but quite counter-intuitive (that is is a paradox). […]
A new mapping from funcoids to reloids
Less 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 the following conjectures: Conjecture $latex (\mathsf{RLD})_X f = \min \left\{ g \in \mathsf{RLD} […]
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 […]
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 $latex f$?