New theorem about core part of funcoids and reloids
Today I’ve proved a new little theorem: Theorem $latex \mathrm{Cor} ( \mathsf{FCD}) g = ( \mathsf{FCD}) \mathrm{Cor}\, g$ for every reloid $latex g$. Conjecture For every funcoid $latex g$ $latex \mathrm{Cor} ( \mathsf{RLD})_{\mathrm{in}} g = ( \mathsf{RLD})_{\mathrm{in}} \mathrm{Cor}\, g$; $latex \mathrm{Cor} ( \mathsf{RLD})_{\mathrm{out}} g = ( \mathsf{RLD})_{\mathrm{out}} \mathrm{Cor}\, g$. See my book.
A new theorem about funcoids and reloids
I’ve added to preprint of my book a new simple theorem: Theorem $latex \mathrm{GR} ( \mathsf{FCD}) g \supseteq \mathrm{GR}\, g$ for every reloid $latex g$. This theorem is now used in my article “Compact funcoids”.
Product of compact funcoids is compact
I have proved (for any products, including infinite products): Product of directly compact funcoids is directly compact. Product of reversely compact funcoids is reversely compact. Product of compact funcoids is compact. The proof is in my draft article and is not yet thoroughly checked for errors.
An attempt to generalize a theorem failed
Using “compactness of funcoids” which I defined earlier, I’ve attempted to generalize the classic general topology theorem that compact topological spaces and uniform spaces bijectively correspond to each other. I’ve resulted with the theorem Theorem Let $latex f$ is a $latex T_1$-separable (the same as $latex T_2$ for symmetric transitive) compact funcoid and $latex g$ […]
Conjecture: Connectedness in proximity spaces
I’ve asked this question at math.StackExchange.com Let $latex \delta$ be a proximity. A set $latex A$ is connected regarding $latex \delta$ iff $latex \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$. Conjecture Set $latex A$ is connected regarding $latex \delta$ iff for […]
On a common generalization of funcoids and reloids
Just a few seconds ago I had an idea how to generalize both funcoids and reloids. Consider a precategory, whose objects are sets product $latex \times$ of filters on sets ranging in morphisms of this category operations $latex \mathrm{dom}$ and $latex \mathrm{im}$ from the morphisms of our precategory to filters on our objects (sets) This […]
Conjecture about funcoids proved
I’ve proved this my conjecture: $latex g \circ f = \bigsqcap \left\{ G \circ F \,|\, F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \right\}$ for every composable funcoids $latex f$ and $latex g$. See my book (in the current draft the theorem 6.65) for a proof.
An error in my book
In the draft of my book there was an error. I’ve corrected it today. Wrong: $latex \forall a, b \in \mathfrak{A}: ( \mathrm{atoms}\, a \sqsubset \mathrm{atoms}\, b \Rightarrow a \subset b)$. Right: $latex \forall a, b \in \mathfrak{A}: ( a \sqsubset b \Rightarrow \mathrm{atoms}\, a \subset \mathrm{atoms}\, b)$. There are the same error in my […]
A filter which cannot be partitioned into ultrafilters
I’ve proved: There exists a filter which cannot be (both weakly and strongly) partitioned into ultrafilters. It is an easy consequence of a lemma proved by Niels Diepeveen (also Karl Kronenfeld has helped me to elaborate the proof). See the preprint of my book.
A negative result on a conjecture
Due my research about singularities the problem formulated in this blog post was solved negatively with help of Alex Ravsky who has found a counter-example. The conjecture was: $latex \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$ is closed under finite intersections. The counter-example follows: $latex f=\{(x,y)\in\mathbb R^2:|x|\le |y| \vee y=0\}$, $latex g=\{(x,y)\in\mathbb R^2:|x|\ge |y| \vee x=0\}$. It is easy […]