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} […]
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 $latex f$?
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.
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 $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{in}} g=(\mathsf{RLD})_{\mathsf{in}}\mathsf{Compl} g$; $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{out}} g=(\mathsf{RLD})_{\mathsf{out}}\mathsf{Compl} g$;.
About “Each regular paratopological group is completely regular” article
In this blog post I consider my attempt to rewrite the article “Each regular paratopological group is completely regular” by Taras Banakh, Alex Ravsky in a more abstract way using my theory of reloids and funcoids. The following is a general comment about reloids and funcoids as defined in my book. If you don’t understand […]
An important conjecture about funcoids. Version 2
This conjecture appeared to be false. Now I propose an alternative conjecture: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is nonempty. […]
An important conjecture about funcoids
Just a few minutes ago I’ve formulated a new important conjecture about funcoids: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is […]
Join of transitive reloids (a conjecture in uniformity theory)
Conjecture Join of a set $latex S$ on the lattice of transitive reloids is the join (on the lattice of reloids) of all compositions of finite sequences of elements of $latex S$. It was expired by theorem 2.2 in “Hans Weber. On lattices of uniformities”. There is a similar conjecture for funcoids (instead of reloids). […]