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.
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.
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 conjecture about filters proved
I have proved this recently formulated conjecture. See my book. Currently it is theorem number 598.
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 $latex X$. […]
Very easy solution of my old conjecture
Like a complete idiot, this took me a few years to disprove my conjecture, despite the proof is quite trivial. Here is the complete solution: Example $latex [S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$, where $latex [S]$ is the complete lattice generated by a strong partition $latex S$ of filter on a set. Proof Consider any infinite set $latex U$ and its […]