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.