I claimed earlier that I partially solved this open problem. Today I solved it completely. The proof is available in this PDF file.
read moreI’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft. Both chapters are very rough draft and present not rigorous proofs but rough ideas.
read moreThe journal European Journal of Pure and Applied Mathematics has accepted my article after a peer review and asked me to send it in their LaTeX format. I had a hyperref trouble with my LaTeX file. So I’ve said them that I …
read moreI am reading the book “Convergence Foundations of Topology” by Szymon Dolecki and Frédéric Mynard. (Well, I am more skimming than reading, I may read it more carefully in the future.) After reading about a half of the book, I tried to…
read moreI (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)….
read moreI 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…
read moreI 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…
read moreThe 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…
read moreI 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.
read moreWhile 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.
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