TITLE: Characterization of Tychonoff spaces in terms of open sets
QUESTION [6 upvotes]: Metrizability and complete regularity are topological properties that are, in a sense, different from the Hausdorff condition because they are not defined purely in the terms of the open sets, but rather using some external object, namely $\mathbb R$. Now a space is metrizable if it has the weak topology induced by a function $d : X \times X \rightarrow \mathbb R$ satisfying the properties of a metric, but metrization theorems tell us that an equivalent condition is that $X$ is regular and has a sigma-discrete basis (and this is purely topological). Is there a similiar characterization of Tychonoff spaces that makes no reference to $\mathbb R$ whatsoever?

REPLY [3 votes]: Suppose that $(X,\mathcal{T})$ is a topological space. Then define a relation $\prec$ on $\mathcal{T}$ by letting $U\prec V$ if and only if $\overline{U}\subseteq V$. Define a relation $\ll$ on $\mathcal{T}$ by letting $U\ll V$ if and only if there is some $(U_{r})_{r\in[0,1]\cap\mathbb{Q}}$ such that $U_{0}=U,U_{1}=V$ and $\overline{U_{r}}\subseteq U_{s}$ whenever $r<s$. Then 

the space $(X,\mathcal{T})$ is regular if and only if for each $U\in\mathcal{T}$, we have $U=\bigcup\{V\in\mathcal{T}|V\prec U\}$, and 
the space $(X,\mathcal{T})$ is completely regular if and only if for each $U\in\mathcal{T}$, we have $U=\bigcup\{V\in\mathcal{T}|V\ll U\}$.

This characterization of regular and completely regular spaces is the way that one typically defines regularity and complete regularity in point-free topology. In fact, in point-free topology it is much more convenient to use this characterization of complete regularity than it is to use the characterization of complete regularity in terms of frame homomorphisms from the real numbers to your frame.
Perhaps the most illuminating characterizations of complete regularity are the following characterizations.
Let $X$ be a topological space. Then the following are equivalent:

$X$ is completely regular.
$X$ has a compatible proximity.
$X$ has a compatible uniformity.

Of course, proximities and uniformities impose additional external structure on topological spaces.