### Directed topological spaces and funcoids

I have researched relations between directed topological spaces and pair of funcoids. Here the first funcoid represents topology and the second one represents direction. Results are mainly negative: Not every directed topological space can be represented as a pair of funcoids. Different…

### 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…

### 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.

### 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…

### 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…