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 […]

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 […]

One more conjecture about provability without axiom of choice

I addition to this conjecture I formulate one more similar conjecture: Conjecture $latex a\setminus^{\ast} b = a\#b$ for arbitrary filters $latex a$ and $latex b$ on a powerset cannot be proved in ZF (without axiom of choice). Notation (where $latex \mathfrak{F}$ is the set of filters on a powerset ordered reverse to set-theoretic inclusion): $latex […]