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 […]
Connectedness of funcoids and reloids – an error corrected
I have corrected some errors in my book about connectedness of funcoids and reloids. In some theorems I replace like $latex S(\mu)$ with $latex S_1(\mu)$ and arbitrary paths with nonzero-length paths. I also discovered (not yet available online) some new results about connected funcoids.
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. […]
ERROR: Section “Micronization” removed
“Micronization” was a thoroughly wrong idea with several errors in the proofs. This section is removed from the book.
Error in my math book
I’ve noticed that the statement “Micronization is always reflexive.” in my math book is erroneous. It led also to some further errors in the section “Micronization”. I am going to correct the errors in near time.
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 new funcoid discovered
It is easy to prove that the equation $latex \langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X$ (for principal filters $latex X$) defines a (unique) funcoid $latex \mathscr{A}$ which I call quasi-atoms funcoid. Note that as it is easy to prove $latex \langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y$ for every set $latex Y$ of ultrafilters. […]