In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\,\to \,C(X)$ is an epimorphism in the category $\mathcal{C}\mathcal{R}$ of commutative rings (with units). We call such an embedding a $\mathcal{C}\mathcal{R}$-epic embedding and we say that $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if every embedding of $X$ is $\mathcal{C}\mathcal{R}$-epic. We continue this investigation. Our most notable result shows that a Lindelöf space $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all $\sigma $-compact spaces and all Lindelöf $P$-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute $\mathcal{C}\mathcal{R}$-epic.