Totally bounded reloids, a generalization of totally bounded uniform spaces
This is a straightforward generalization of the customary definition of totally bounded sets on uniform spaces: Definition Reloid $latex f$ is totally bounded iff for every $latex E \in \mathrm{GR}\, f$ there exists a finite cover $latex S$ of $latex \mathrm{Ob}\, f$ such that $latex \forall A \in S : A \times A \subseteq E$. […]
“Simple product”, a new kind of product of funcoids
Today I’ve discovered a new kind of product of funcoids which I call “simple product”. It is defined by the formulas $latex \left\langle \prod^{(S)}f \right\rangle x = \lambda i \in \mathrm{dom}\, f: \langle f_i \rangle x_i$ and $latex \left\langle \left( \prod^{(S)}f \right)^{-1} \right\rangle y = \lambda i \in \mathrm{dom}\, f:\langle f_i^{-1} \rangle y_i$. Please read […]
A partial proof of “Partitioning a filter into ultrafilters” conjecture
I’ve put a partial partial proof of “Every filter on a set can be strongly partitioned into ultrafilters” conjecture at PlanetMath. Please collaborate in solving this conjecture.
Online listing of research and other projects to participate
I’ve created a site where anyone may list his projects and anyone may mark which projects he is going to participate. Projects are organized into a tree. The site supports LaTeX and has “Mathematics” section: http://theses.portonvictor.org/node/2 I have posted several pages on my math research project: http://theses.portonvictor.org/node/4 – feel free to add yours also. The […]
Definition of subatomic projection of funcoids
I have proved that for every funcoid $latex f:\prod A\rightarrow\prod B$ (where $latex A$ and $latex B$ are indexed families of sets) there exists a funcoid $latex \mathrm{Pr}^{(A)}_k f$ (subatomic projection) defined by the formula: $latex \mathcal{X} \mathrel{\left[ \Pr^{\left( A \right)}_k f \right]} \mathcal{Y} \Leftrightarrow \\ \prod^{\mathsf{RLD}}_{i \in \mathrm{dom}\, A} \left( \left\{ \begin{array}{ll} 1^{\mathfrak{F} \left( […]
A breakthrough and getting stuck
I had a very great idea in the field of general topology, what I now call funcoids. Years of my research of funcoids culminated me writing a research monograph about funcoids and related stuff. But after I’ve finished this monograph and submitted it to a publisher, I realize that I get stuck. I have no […]
A new section in my research monograph
I added the definition and properties of “second reloidal product” (the definition was inspired by Tychonoff product of topological spaces) to my research monograph “Algebraic General Topology. Volume 1”. See the subsection “Second reloidal product” in the section “Multireloids”.
Rough idea: Operations on singularities and their application to general relativity
I discovered a math theory which (among other things) gives an alternate interpretation of the equations of general relativity (something like to replacing real numbers with complex numbers in a quadratic equation). This theory is a theory of limits in points of singularities and properties of singularities based on my theory of funcoids (a new […]
A new simple proposition about generalized limits
I’ve added the following almost trivial proposition to the draft of my book “Algebraic General Topology. Volume 1”: Proposition $latex \tau \left( y \right) = \mathrm{xlim}\, \left( \left\langle \mu \right\rangle^{\ast} \left\{ x \right\} \times^{\mathsf{FCD}} \uparrow^{\mathrm{Base}\, \left( \mathrm{dom}\, \nu \right)} \left\{ y \right\} \right)$ (for every $latex x$). Informally: Every $latex \tau \left( y \right)$ is […]
Meta-singular numbers
I’ve defined (well, vaguely defined) what I call “meta-singular numbers”. These can be used to describe values of a function in a singularity. See this PlanetMath article.