Let $X$ be a uniformizable topological space and $\mathcal{U}_\Delta$ consist of all neighbourhoods of the diagonal $\Delta_X$ of $X$.
We say uniformity $\mathcal{U}$ on $X$ is admissible if it induces the same topology as that of $X$.
If $\mathcal{U}_\Delta$ is a uniformity on $X$, then it's an admissible uniformity. The only way in which $\mathcal{U}_\Delta$ may fail to be a uniformity, is if there exists a neighbourhood of the diagonal $U$ such that there's no neighbourhood of the diagonal $V$ with $V\circ V\subseteq U$. If $X$ is paracompact Hausdorff then $\mathcal{U}_\Delta$ is a uniformity on $X$.
On the other hand consider the uniformity $\mathcal{U}_F$ consisting of all neighbourhoods $U$ of the diagonal $\Delta_X$ such that there exists a sequence $U_n$ of neighbourhoods of $\Delta_X$ with $U_1 = U$ and $U_{n+1}\circ U_{n+1}\subseteq U_n$. This is called the fine uniformity on $X$, and its the largest admissible uniformity on $X$.
What is an example of a uniformizable space $X$ such that $\mathcal{U}_\Delta\neq \mathcal{U}_F$, or equivalently, $\mathcal{U}_\Delta$ is not a uniformity?
