The category of continuous maps between endofuncoids has small products
I have proved that Theorem The category of continuous maps between endofuncoids has small products. See my draft article for a proof.
I’ve fixed the error in my proof
I have quickly corrected the error in my proof of an important theorem. Now it is even more beautiful.
Error in my proof
That proof which I claimed in this blog post is with an error: I have messed product of objects and product of morphisms. Now I desperately attempt to repair the proof.
Direct product in the category of continuous maps between endofuncoids
I released a rough draft of my article Direct product in the category of continuous maps between endofuncoids. This (among other) solves the problem I proposed in this blog post. Previously I have said that my research got stuck. Now I see how to continue it! I am again blessed.
Conjecture solved
I’ve published in my book’s preprint a theorem (currently numbered 6.100) which solves a former conjecture. Theorem $latex g \circ \left( \bigsqcup R \right) = \bigsqcup \left\{ g \circ f \,|\, g \in R \right\} = \bigsqcup \langle g \circ \rangle R$ if $latex g$ is a complete funcoid. My shame, I have earlier overlooked […]
Open mappings between endo-funcoids
Let $latex \mu$ and $latex \nu$ are endofuncoids and $latex f$ is a funcoid from $latex \mathrm{Ob}\,\mu$ to $latex \mathrm{Ob}\,\nu$. Then we can generalize Bourbaki’s notion of open mapping between topological spaces (that is a mapping for which images of open sets are open) by the following formula (where $latex x$ is a variable which […]
The (candidate) construction of direct product in the category of continuous maps between endo-funcoids
Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids. Remind from my book that morphisms $latex f: A\rightarrow B$ of this category are defined by the formula $latex f\circ A\sqsubseteq B\circ f$ (here and below by abuse of notation I equate functions with corresponding principal funcoids). Let $latex F_0, F_1$ are […]
My fool topology study
I am (re)reading Bourbaki “General Topology” (in Russian language). Despite I am a general topologist, I have never had a systematic general topology study. I think now I should fill this hole. Maybe after reading Bourbaki I will return to Johnstone “Stone Spaces” which I weakly if at all understand.
Pointfree funcoids as a generalization of frames/locales
I’ve put online my rough partial draft of the theory of bijective correspondence between frames/locales and certain pointfree funcoids. Pointfree funcoids are a massive generalization of locales and frames: They not only don’t require the lattice of filters to be boolean but these can be even not lattices of filters at all but just arbitrary […]
My study of pointfree topology
I have read The point of pointless topology today and am going to study the book Johnstone “Stone Spaces” which I purchased maybe a year or two ago. The purpose of this study is to integrate others’ pointless topology with my theory of pointfree funcoids. From my earlier comment on this blog: It seems that […]