Mappings between endofuncoids and topological spaces
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 induced by a topological space. The formula is $latex f\mapsto 1\sqcup\mathrm{Compl}\, f\sqcup(\mathrm{Compl}\, f)^2\sqcup […]
Expressing limits as implications
I 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 arbitrary ultrafilter. The section also contains another, a little less obvious theorem. There […]
A new partial result about products of filters [ERROR!]
Below 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 filters $latex \mathcal{A}$, $latex \mathcal{B}$, $latex \mathcal{C}$. Currently the proof is located in this […]
Join of two connected (regarding a funcoid) filters, whose meet is proper, is connected
I 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 going to move it to the book in the future.)
More on connectedness of filters
I 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 that I order filters reversely to set-theoretic inclusion.) This is now an important […]
Counting sides of a surface topologically
I 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 closed disk). Currently these definitions and questions are presented in the file addons.pdf. […]
Theory of filters is FINISHED!
I 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 the attached image: Note that as it seems nobody before me researched filters […]
A new conjecture about filters
Let $latex \mathfrak{F}(S)$ 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 $latex Z(X)$ denotes the set of complemented elements of the lattice $latex X$. […]
A counterexample to my conjecture
I’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$ The counterexample is $latex f=a\times^{\mathsf{FCD}} \{p\}$ and $latex g=\{p\}\times^{\mathsf{FCD}}a$, $latex H=1$ where $latex […]
A conjecture about funcoids on real numbers disproved
I 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. Note that I have not yet proved $latex \lvert \mathbb{R} \rvert_{>} \neq \lvert […]