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 […]
My further study plans
I remind that I am not a professional mathematician. Nevertheless I have written research monograph “Algebraic General Topology. Volume 1”. Yesterday I have asked on MathOverflow how to characterize a poset of all filters on a set. From the answer: the posets isomorphic to lattices of filters on a set are precisely the atomic compact […]
A new math problem about funcoids
Just a few seconds ago I realized that I have never considered and and even never formulated the following problem: Explicitly describe the set of complemented funcoids. Note that not all principal funcoids are complemented. For example see my book for a proof that the identity funcoid on some set is not complemented.
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 […]