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 […]
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 […]
Conjecture: Distributivity of a lattice of funcoids is not provable without axiom of choice
Conjecture Distributivity of the lattice $latex \mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $latex A$ and $latex B$) is not provable in ZF (without axiom of choice). It is a remarkable conjecture, because it establishes connection between logic and a purely algebraic equation. I have come to this conjecture in the following way: My proof that […]
Definition of order of pointfree funcoids changed in my book
In this blog post I announced that I am going to change the definition of order of pointfree funcoids in my book. Now in the last preprint the changes are done.
Changed the definition of order of pointfree funcoids
In my preprint I defined pre-order of pointfree funcoids by the formula $latex f\sqsubseteq g \Leftrightarrow [f]\subseteq[g]$. Sadly this does not define a poset, but only a pre-order. Recently I’ve found an other (non-equivalent) definition of an order on pointfree funcoids, this time this is a partial order not just a pre-order: $latex f \sqsubseteq […]
Pointfree funcoid induced by a locale or frame?
I have shown in my research monograph that topological (even pre-topological) spaces are essentially (via an isomorphism) a special case of endo-funcoids. It was natural to suppose that locales or frames induce pointfree funcoids, in a similar way. But I just spent a few minutes on defining the pointfree funcoid corresponding to a locale or […]