Partial order funcoids and reloids
Partial order funcoids and reloids formalize such things as “infinitely small” step rotating a circle counter-clockwise. This is “locally” a partial order as every two nearby “small” sets (where we can define “small” for example as having the diameter (measuring along the circle) less than $latex \pi$) are ordered: which is before in the order […]
The history of discovery of funcoids
In my book I introduce funcoids as a generalization of proximity spaces. This is the most natural way to introduce funcoids, but it was not the actual way I’ve discovered them. The first thing discovered equivalent to funcoids was a function $latex \Delta$ (generalizing a topological space) which I defined to get a set as […]
Algebraic General Topology presentation – new version
The PDF Slides about Algebraic General Topology were updated to match newer notation used in my book. Use these slides to quickly familiarize yourself with my theory. I’ve removed altogether the notion of filter object, instead using a new different notation of lattice operators.
A mathematical theory of singularities!
I present my mathematical theory of singularities. It may probably have applications in general relativity and other physics. The definitions are presented in this short draft article. Before reading this article I recommend to skim through my research monograph (in the field of general topology), because the above mentioned article uses concepts defined in my […]
Two theorems about totally bounded images of a totally bounded reloid
I’ve added to my book two following theorems (formerly conjectures). Theorem Let $latex \mu$ and $latex \nu$ are endoreloids. Let $latex f$ is a principal $latex \mathrm{C}’ ( \mu; \nu)$ continuous, monovalued, surjective reloid. Then if $latex \mu$ is $latex \beta$-totally bounded then $latex \nu$ is also $latex \beta$-totally bounded. Theorem Let $latex \mu$ and […]
Alternate definition of quasi-uniform and quasi-metric spaces
Reloid is a triple $latex {( A ; B ; F)}&fg=000000$ where $latex {A}&fg=000000$, $latex {B}&fg=000000$ are sets and $latex {F}&fg=000000$ is a filter on their cartesian product $latex {A \times B}&fg=000000$. Endoreloid is reloid with the same $latex {A}&fg=000000$ and $latex {B}&fg=000000$. Uniform space is essentially a special case of an endoreloid. The reverse […]
New chapter in my research monograph
I’ve added new chapter 11 “Total boundness of reloids” to my book “Algebraic General Topology. Volume 1”. It expresses several kinds of boundness of reloids, which are however the same total boundness in the special case of uniform spaces.
Change of terminology
I realized that the terms “discrete funcoid” and “discrete reloid” conflict with conventional usage of “discrete topology” and “discrete uniformity”. Thus I have renamed them into “principal funcoid” and “principal reloid”. See my research monograph.
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 […]