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July 23, 2016 / porton

Normality of a quasi-uniform space on a topology is determined by the proximity induced by the quasi-uniform space

First a prelude:

Taras Banakh, Alex Ravsky¬†“Each regular paratopological group is completely regular” solved a 60 year old open problem.

Taras Banakh introduces what he call normal uniformities (don’t confuse with normal topologies).

My new result, proved with advanced funcoids theory (and never tried to prove it with basic general topology): Whether a uniformity on a topology is normal is determined by the proximity induced by the uniformity. (Moreover I expressed it as an explicit algebraic formula in terms of funcoids: \nu\circ\nu^{-1}\sqsubseteq\nu^{-1}\circ\mu, where \mu is the proximity induced by the quasi-uniformity and \nu is the topological space).

July 22, 2016 / porton

Virtual mathematics conferences opened!

I have just created a new wiki Web site, which is a virtual math conference,
just like a real math meeting but running all the time (not say once per two
years).

https://conference.portonvictor.org

Please spread the word that we have a new kind of math conference.

Please post references to your articles, videos, slides, etc.

July 21, 2016 / porton

New proof of Urysohn’s lemma

I present a new proof of Urysohn’s lemma. Well, not quite: my proof is dependent on an unproved conjecture.

Currently my proof is present in this PDF file.

The proof uses theory of funcoids.

July 20, 2016 / porton

A math question

What are necessary and sufficient conditions for \mathrm{up}\, f to be a filter for a funcoid f?

July 20, 2016 / porton

New easy theorem and its consequence

I’ve added to my book a new easy to prove theorem and its corollary:

Theorem If f is a (co-)complete funcoid then \mathrm{up}\, f is a filter.

Corollary

  1. If f is a (co-)complete funcoid then \mathrm{up}\, f = \mathrm{up} (\mathsf{RLD})_{\mathrm{out}} f.
  2. If f is a (co-)complete reloid then \mathrm{up}\, f = \mathrm{up}\, (\mathsf{FCD}) f.
July 12, 2016 / porton

A new conjecture

While writing my book I forgot to settle the following conjecture:

Conjecture \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F for every composable funcoids f and g.

Note that the similar statement about reloids is quite obvious.

July 10, 2016 / porton

A new (but easy to prove) theorem in my research book:

Theorem Let \mu and \nu be endomorphisms of some partially ordered dagger precategory and f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu) be a monovalued, entirely defined morphism. Then f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).

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