Given topological spaces $X$ and $C$ we call $C$ a **coordinate space** for $X$ to mean that every open set $U \subset X$ is of the form $f^{-1}(V)$ for some open $V \subset C$ and continuous $f \colon X \to C$.

More generally if $\mathscr X$ is a class of spaces we call $C$ a **coordinate space** for $\mathscr X$ to mean it is a coordinate space for every member of $\mathscr X$.

Two familiar examples of coordinate spaces:

The Seirpinski space $\{0,1\}$ with exactly one closed point and one open point is a coordinate space for the class of all topological spaces.

The closed unit interval is a coordinate space for the class of all completely-regular spaces. Moreover the closed unit interval is itself completely-regular.

But what about the class of all Hausdorff spaces? Obviously the Seirpinski space is still a coordinate space here. But what if we demand the coordinate space itself be Hausdorff -- as with example 2?

I would reckon such a space does not exist but have no idea how to prove that?

Does this notion already have a name and has the question been addressed before?