Fine uniformity on a set

Question

I came across the term fine uniformity while reading a paper. I want to know

What is meant by fine uniformity on a set? Is this the largest uniformity that can be defined on it? That is, uniformity containning diagonal of the set?

Answer 1

A topological space is uniformizable if and only if it is completely regular. One can show that, on a completely regular space, there is a finest uniformity compatible with the topology, called the fine uniformity.

The fine uniformity is generated by all open neighborhoods $D$ of the diagonal of $X \times X$ (with the product topology) such that there exists a sequence $(D_n)_{n \geqslant 0}$ of open neighborhoods of the diagonal such that $D = D_0$ and $D_{n+1}\circ D_{n+1}\subseteq D_n$ for all $n$. See the wikipedia entry uniformizable space for more details.

In contrast, every topological space is quasi-uniformizable (the definition of a quasi-uniformity is the same as that of a uniformity, except that the inverse of an entourage is not required to be an entourage).