In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$. The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), but it may fail to have a bounded linear right inverse (an "extension operator"). Equivalently, $\text{ker}R_Y=\{f\in C(X): f_{|Y}=0\}$ may fail to be complemented in $C(X)$.

The possibly simpler counterexample is mentioned here, with $X:=\beta\mathbb N$ and $Y:=\beta\mathbb N\setminus\mathbb N$. Notably, here the closed set $Y$ is not metrisable. In fact (Luther and Martin, 1974) a metrisable $G_\delta$ subset $Y$ of $X$ does admit an extension operator --this generalises the classical Borsuk-Dugundji theorem, where $X$ is a metric space.

Metrisability of $Y$ is certainly not necessary, in general (e.g. in the quite trivial case of a clopen subset $Y$). Being a $G_\delta$ subset, that is, the zero set of a continuous function on $X$, is also not necessary, in general (e.g. in the quite trivial case of $Y=\{x_0\}$, where $x_0\in\ X$ does not have a countable basis of neighborhoods).

Question. Can we weaken the metrisability condition from Luther-Martin theorem, that is, what more general conditions on a closed set $Y$ ensure the existence of an extensor?

Rmk. An extensor $E:C(Y)\to C(X)$ is always of the form $Ef(x)= \langle Px,f\rangle$, where $P:X\to C(Y)^*$ is continuous w.r.to the $w^*$ topology of $C(Y)^*$ and $Py=\delta_y$ for all $y\in Y$.