A counterexample to my recent conjecture
After proposing this conjecture I quickly found a counterexample: $latex S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $latex f$ is the usual Kuratowski closure for $latex \mathbb{R}$.
New conjecture about funcoids
Conjecture $latex \langle f \rangle \bigsqcup S = \bigsqcup_{\mathcal{X} \in S} \langle f \rangle \mathcal{X}$ if $latex S$ is a totally ordered (generalize for a filter base) set of filters (or at least set of sets).
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 […]
Offtopic: Formalized Gospel theology
This is partly an offtopic post in my math blog. It seems likely that I discovered a category in which such objects as the Father and the Son from the Gospel appear. I am not sure I really discovered God, but this seems likely. Consider a category (there seems to be multiple ways to add […]
My old files related with math logic
In 2005 year I put online some math articles related with formulas and math logic (despite I am not a professional logician). In 2005 I like a crackpot thought that I discovered a completely new math method replacing axiomatic method. This was a huge error (my skipped proof was just wrong). After that the files […]
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.