Sets of integral curves described in topological terms
I (with some twist) described the set of $latex C^1$ integral curves for a given vector field in purely topological terms (well, I describe it not in terms of topological spaces, but in terms of funcoids, more abstract objects than topological spaces). From this PDF file: Theorem $latex f$ is a reparametrized integral curve for […]
Filter rebase generalized
I have re-defined filter rebase. Now it is defined for arbitrary filter $latex \mathcal{A}$ on some set $latex \mathrm{Base}(\mathcal{A})$ and arbitrary set $latex A$. The new definition is: $latex \mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\cap A\subseteq X \}$. It is shown that for the special case of $latex \forall X\in\mathcal{A}:X\subseteq A$ the […]
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 pairs of a topological space and its subfuncoid may generate the same directed […]
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} […]