Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity.

To construct Fine uniformities, Let the family $\{\mathcal{V}_i:i\in I\}$ of all compatible uniformities on $X$ and generates the initial uniformity $\mathcal U$ w.r.t. $id_i:X\to(X,\mathcal{V}_i)$ on $X$. therefore, $\mathcal U$ is finer than each $\mathcal{V}_i$ for $i\in I$. Let us note that, completely regularity is used to show that $\{\mathcal{V}_i:i\in I\}\not=\emptyset$, since $X$ is a completely regular(uniformizable) space and have a compatible uniformity.

Exercise 36H. from Willard, General Topology says that,

The fine uniformity $\mathcal D_F$ on a uniformizable space is the uniformity having for a base $\beta$, the open sets $D\supset\Delta$ such that for some sequence $D_1, D_2,...$ of open sets containing $\Delta$, $D_n\circ D_n\subset D_{n-1}$ for all $n$ and $D_1= D$.

Is $\mathcal U$ and $\mathcal D_F$ exactly the same uniformity on a uniformizable space $X$? If the answer is yes, then how we can construct a base $\beta$ for $\mathcal U$ ?

I am trying to solve this question. $\beta$ is a base for a uniformity on $X$ and the uniformity $\mathcal D_F$ generated by $\beta$ is finer than $\mathcal U$. now it is enough to show that, $\mathcal D_F\subset \mathcal U$. If the $\mathcal D_F$ is compatible with the topology of $X$, then $\mathcal D_F\subset \mathcal U$, since $\mathcal U$ is the finest uniformity on $X$ compatible with the topology of $X$.

Now my problem is to show that $\mathcal D_F$ is compatible with the topology of $X$.

I asked this question 3 months ago [here] but still i can't find the answer. thanks in advice

• The first quote is from where? Why doesn't 36H. answer your question? It's a specific base for the fine uniformity, is it not? Or are you looking for a complete proof. – Henno Brandsma Apr 25 '16 at 17:43
• mathoverflow.net/a/227702/2060 does not answer the question (in another way?) – Henno Brandsma Apr 25 '16 at 18:05
• @HennoBrandsma I cant find the same definition of the fine uniformity and this question is to show that this two kind of definitions are equal. The first quote is concluded from this question and Chapter 0 of Roelcke & Dierolf, Uniform Structures on Topological Groups and their Quotients. A complete proof can clear all the problems. – M.A. Apr 25 '16 at 18:08
• The answer by Dominic I linked to is the "explicit" construction of the supremum of all compatible uniformities. So by definition almost it induces the fine uniformity (defined as the finest compatible one). – Henno Brandsma Apr 25 '16 at 18:10
• Also see aw.twi.tudelft.nl/~hart/37/publications/the_papers/ency/… (there is a small section on the fine uniformity). Apparently the covers with a locally finite refinement of co-zero sets are exactly the covers in the fine uniformity. – Henno Brandsma Apr 25 '16 at 18:50