I started research of mappings between endofuncoids and topological spaces. Currently the draft is located in volume 2 draft of my online book. I define mappings back and forth between endofuncoids and topologies. The main result is a representation of an endofuncoid…

read moreI have added to my book section “Expressing limits as implications”. The main (easy to prove) theorem basically states that $latex \lim_{x\to\alpha} f(x) = \beta$ when $latex x\to\alpha$ implies $latex f(x)\to\beta$. Here $latex x$ can be taken an arbitrary filter or just…

read moreBelow contains an error. Trying to calculate $latex (\mathcal{B} \times^{\mathsf{RLD}}_F \mathcal{C}) \circ (\mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B})$, I’ve proved (not yet quite thoroughly checked for errors) the following partial result: Proposition $latex (\mathcal{B} \times^{\mathsf{RLD}}_F \mathcal{C}) \circ (\mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B}) \neq \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{C}$ for some proper…

read moreI have proved that join of two connected (regarding a funcoid) filters, whose meet is proper, is connected. (I remind that in my texts filters are ordered reverse set-theoretic inclusion.) The not so complex proof is available in the file addons.pdf. (I am…

read moreI added more on connectedness of filters to the file addons.pdf (to be integrated into the book later). It is a rough incomplete draft. Particularly the proof, that the join of two connected filters with proper meet is connected, is not complete. (Remember…

read moreI have defined sides of a surface (represented by such things as a set in a topological space) purely topologically. I also gave two (possible non-equivalent) definitions of special points of a surface (such “singularities” as points of the border of a…

read moreI have almost finished developing theory of filters on posets (not including cardinality issues, maps between filters, and maybe specifics of ultrafilters). Yeah, it is finished! I have completely developed a field of math. Well, there remains yet some informal problems, see…

read moreLet $latex \mathfrak{F}(S)$ denotes the set of filters on a poset $latex S$, ordered reversely to set theoretic inclusion of filters. Let $latex Da$ for a lattice element $latex a$ denote its sublattice $latex \{ x \mid x \leq a \}$. Let…

read moreI’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$…

read moreI proved that $latex \lvert \mathbb{R} \rvert_{\geq} \neq \lvert \mathbb{R} \rvert \sqcap \geq$ and so disproved one of my conjectures. The proof is currently available in the section “Some inequalities” of this PDF file. The proof isn’t yet thoroughly checked for errors….

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