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.
Please nominate me for Breakthrough Prize in mathematics
Please read my math research and decide if it is worth the prize. If you consider my research worth the prize, please nominate me. Nominations for the Breakthrough Prize and New Horizons Prize in mathematics are now open. The Breakthrough Prize in Mathematics is a $3,000,000 prize for transformative breakthrough(s) in mathematics, with special attention […]
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.