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 to be a quasi-uniform space and is the corresponding quasi-proximity.

is the closure operator corresponding to a funcoid . I also denote the image of a function as .

I will also denote the *interior* funcoid for a co-complete funcoid (for the special case if is a topological space 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 is a quasi-uniform space) an endo-reloid on a set is *normal* when for every entourage of and every set .

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

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

The further step fails because in general .

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

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