Explicit description of a left adjoint to the forgetful functor from Unif to Top

Question

Let $\newcommand\Unif{\mathrm{Unif}}\Unif$ denote the category of uniform spaces and uniformly continuous functions and $\newcommand\Top{\mathrm{Top}}\Top$ the category of topological spaces and continuous functions. We have a forgetful functor $\Unif\to\Top$ which sends every uniform space to the topological spaces whose topology is induced by the uniformity.

Since the forgeful functor $\Unif\to\Top$ preserves initial sources, it has a left adjoint $\Top\to\Unif$. Thus to every topological space is associated a canonical uniformity compatible with its topology.

Is possible to give an explicit costruction of this uniformity?

I try to consider as set of entourages the set of the neighbourhoods of the diagonal, but I'm failed to show that every entourage $U$ contains $V\circ V$ form some entourage $V$.

Answer 1

It does not follow from the existence of a left adjoint $F\colon \mathsf{Top}\to \mathsf{Unif}$ that every topological space has a canonical uniformity compatible with its topology. There's no reason to expect that $F(X)$ is homeomorphic to $X$.

It turns out that a topological space admits a uniformity compatible with its topology if and only if it is completely regular. A completely regular space admits a finest uniformity compatible with its topology, called the fine uniformity, and the functor $\mathsf{CReg}\to \mathsf{Unif}$ which assigns to each completely regular space its fine uniformity is left adjoint to the forgetful functor $\mathsf{Unif}\to \mathsf{CReg}$. See the wikipedia article on Uniformizable space. See here for a concrete description of the fine uniformity in terms of covers (thanks to Henno Brandsma for the reference and nice answer).

To get the desired left adjoint $\mathsf{Top}\to \mathsf{Unif}$, one just needs to compose the fine uniformity functor with the complete regularization functor $\mathsf{Top}\to \mathsf{CReg}$ (which is left adjoint to the forgetful functor $\mathsf{CReg}\to \mathsf{Top}$). This functor takes a topological space $X$ and replaces the topology on $X$ by the one generated by the cozero sets (i.e. with basis consisting of the sets $Y_f = \{x\in X\mid f(x)\neq 0\}$, for all continuous functions $f\colon X\to \mathbb{R}$). This is described in the wikipedia article on Tychonoff space.