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June 23, 2016 / porton

About “Each regular paratopological group is completely regular” article

In this blog post I consider my attempt to rewrite the article “Each regular paratopological group is completely regular” by Taras Banakh, Alex Ravsky in a more abstract way using my theory of reloids and funcoids.

The following is a general comment about reloids and funcoids as defined in my book. If you don’t understand them, restrict your mind to the special case {f} to be a quasi-uniform space and {(\mathsf{FCD}) f} is the corresponding quasi-proximity.

{\langle f \rangle^{\ast}} is the closure operator corresponding to a funcoid {f}. I also denote the image {\mathscr{P} X \rightarrow \mathscr{P} Y} of a function {f : X \rightarrow Y} as {\langle f \rangle^{\ast}}.

I will also denote {f^{\circ}} the interior funcoid for a co-complete funcoid {f} (for the special case if {f} is a topological space {f^{\circ}} is the interior operator of this space). It is defined in the file addons.pdf (not yet in my book).

By definition (slightly generalizing the special case if {f} is a quasi-uniform space) an endo-reloid {f} on a set {U} is normal when {\langle (\mathsf{FCD}) f \rangle^{\ast} A \sqsubseteq \langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \langle (\mathsf{FCD}) f \rangle^{\ast} \langle F \rangle^{\ast} A} for every entourage {F \in \mathrm{up}\, f} of {f} and every set {A \subseteq U}.

Then it appear “obvious” that this definition of normality is equivalent to the formula:

\displaystyle (\mathsf{FCD}) f \sqsubseteq ((\mathsf{FCD}) f)^{\circ} \circ (\mathsf{FCD}) f \circ (\mathsf{FCD}) f.

However, I have failed to prove it. Here is my attempt

\langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \langle (\mathsf{FCD}) f \rangle^{\ast} \langle (\mathsf{FCD}) f \rangle^{\ast} A = \\ \langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \langle (\mathsf{FCD}) f \rangle^{\ast} \bigsqcap_{F \in \mathrm{up} f} \langle F \rangle^{\ast} A = \\ \langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \bigsqcap_{F \in \mathrm{up} f} \langle (\mathsf{FCD}) f \rangle^{\ast} \langle F \rangle^{\ast} A = ? ?

The further step fails because in general {\langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \bigsqcap_{F \in \mathrm{up} f} S \neq \bigsqcap_{F \in \mathrm{up} f} \langle \langle ((\mathsf{FCD}) f)^{\circ} \rangle^{\ast} \rangle^{\ast} S}.

So as now my attempt has failed. Please give me advice how to overcome this shortcoming of my theory.

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