I have claimed that I have proved this theorem: Theorem Let $latex f$ is a $latex T_1$-separable (the same as $latex T_2$ for symmetric transitive) compact funcoid and $latex g$ is an reflexive, symmetric, and transitive endoreloid such that $latex ( \mathsf{FCD})…
read moreToday I’ve proved a new little theorem: Theorem $latex \mathrm{Cor} ( \mathsf{FCD}) g = ( \mathsf{FCD}) \mathrm{Cor}\, g$ for every reloid $latex g$. Conjecture For every funcoid $latex g$ $latex \mathrm{Cor} ( \mathsf{RLD})_{\mathrm{in}} g = ( \mathsf{RLD})_{\mathrm{in}} \mathrm{Cor}\, g$; $latex \mathrm{Cor} (…
read moreI’ve added to preprint of my book a new simple theorem: Theorem $latex \mathrm{GR} ( \mathsf{FCD}) g \supseteq \mathrm{GR}\, g$ for every reloid $latex g$. This theorem is now used in my article “Compact funcoids”.
read moreI have proved (for any products, including infinite products): Product of directly compact funcoids is directly compact. Product of reversely compact funcoids is reversely compact. Product of compact funcoids is compact. The proof is in my draft article and is not yet…
read moreUsing “compactness of funcoids” which I defined earlier, I’ve attempted to generalize the classic general topology theorem that compact topological spaces and uniform spaces bijectively correspond to each other. I’ve resulted with the theorem Theorem Let $latex f$ is a $latex T_1$-separable…
read moreI’ve asked this question at math.StackExchange.com Let $latex \delta$ be a proximity. A set $latex A$ is connected regarding $latex \delta$ iff $latex \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta}…
read moreJust a few seconds ago I had an idea how to generalize both funcoids and reloids. Consider a precategory, whose objects are sets product $latex \times$ of filters on sets ranging in morphisms of this category operations $latex \mathrm{dom}$ and $latex \mathrm{im}$…
read moreI’ve proved this my conjecture: $latex g \circ f = \bigsqcap \left\{ G \circ F \,|\, F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \right\}$ for every composable funcoids $latex f$ and $latex g$. See my book (in the current draft the…
read moreIn the draft of my book there was an error. I’ve corrected it today. Wrong: $latex \forall a, b \in \mathfrak{A}: ( \mathrm{atoms}\, a \sqsubset \mathrm{atoms}\, b \Rightarrow a \subset b)$. Right: $latex \forall a, b \in \mathfrak{A}: ( a \sqsubset b…
read moreI’ve proved: There exists a filter which cannot be (both weakly and strongly) partitioned into ultrafilters. It is an easy consequence of a lemma proved by Niels Diepeveen (also Karl Kronenfeld has helped me to elaborate the proof). See the preprint of…
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