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.

We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\,\to \,Y$, the induced map $C(Y)\,\to \,C(X)$ is an epimorphism of rings. Such spaces are called absolute $\mathcal{C}\mathcal{R}$-epic. The simplest examples of absolute $\mathcal{C}\mathcal{R}$-epic spaces are $\sigma $-compact locally compact spaces and Lindelöf $P$-spaces. We show that absolute $\mathcal{C}\mathcal{R}$-epic first countable spaces must be locally compact.

However, a “bad” class of absolute $\mathcal{C}\mathcal{R}$-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute $\mathcal{C}\mathcal{R}$-epic abound, and some are presented.

Some theorems on the existence of continuous real-valued functions on a topological space (for example, insertion, extension, and separation theorems) can be proved without involving uncountable unions of open sets. In particular, it is shown that well-known characterizations of normality (for example the Katětov-Tong insertion theorem, the Tietze extension theorem, Urysohn's lemma) are characterizations of normal σ-rings. Likewise, similar theorems about extremally disconnected spaces are true for σ-rings of a certain type. This σ-ring approach leads to general results on the existence of functions of class α.