Is the fine uniformity generated by all continuous pseudometrics?

Question

Given any topological space $(X,\tau)$, it seems to me that the uniformity generated by all continuous pseudometrics $d:(X,\tau)\times (X,\tau)\to [0,\infty)$ is the fine uniformity of the associated completely regular topology of $\tau$. This would yield an easy desciption of the left adjoint of the functor assigning to a uniform space the generated topological space (a quick search in the web only yields such a left adjoint as the composition of the adjoints of CompReg $\to$ Top and Unif $\to$ CompReg).

Is this correct and do you have a reference?

Answer 1

It is true, and I learned of many of the details in the somewhat obscure paper

D. Thampuran, On Completely Regular Spaces, Portugaliae Mathematica *33, (1974).

Here is the basic idea of how things work.

The full subcategory $CReg$ on the completely regular spaces is bireflective in $Top$. The reflector sends a space $(X,\tau)$ to its completely regular modification $cr(X)$, which is the underlying set $X$ given the topology generated by all cozero sets of $(X,\tau)$.

On the other hand, there is a descrition of $cr(X)$ in terms of continuous pseudometrics. First observe the following.

Lemma: A pseudometric $d:X\times X\rightarrow[0,\infty)$ is $\tau$-continuous if and only if the topology generated by $d$ is weaker than $\tau$.

Thus under $\tau$ in the lattice of topologies on $X$ is a family of pseudometric topologies. The supremum of these topologies need not be pseudometric, but will of course be completely regular. This is true because each pseudometric topology is completely regular, and the supremum of any family of completely regular topologies is completely regular.

Lemma: A set $U\subseteq X$ is a cozero set if and only if it is an open subset of some weaker topology on $X$ which is generated by a continuous pseudometric.

Putting these observations together yields the following.

Corollary: Let $\mathcal{D}$ be the set of all continuous pseudometrics on $(X,\tau)$. Then the completely regular modification $cr(X)$ is the coarsets topology making each member of $\mathcal{D}$ continuous.

So now the affirmiative answer of Jochen's query is clear. The fine uniformity on a completely regular space $(X,\tau)$ is the supremum (in the lattice of uniformities) of all uniformities on $X$ which are compatible with $\tau$. This uniformity is generated by the family of all continuous pseudometics on $(X,\tau)$, a fact which appears as exercise 8.1.C.c in Engelking's General Topology (pg.437 of my edition). (Engelking works only with separated uniformities, but none of the details of the exercise require this.)

On the other hand, by the above discussion, if $(X,\tau)$ is not completely regular, then its family of continuous pseudometrics describes its completely regular modification. Thus the composite functor $$Top\rightarrow CReg\rightarrow Unif$$ is indeed described in the single process given in Jochen's question.