### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### 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 $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$…

### 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….