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…
read moreBelow 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…
read moreI 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…
read moreI 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…
read moreI 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…
read moreI 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…
read more“Micronization” was a thoroughly wrong idea with several errors in the proofs. This section is removed from the book.
read moreI’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.
read moreI’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$…
read moreIt 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…
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