Ends, tangles and critical vertex sets

We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_{\Gamma}$ is a quotient of Diestel's (denoted by $\vert G\vert_{\Theta}$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of $\vert G\vert_{\Theta}$ and our construction of $\vert G\vert_{\Gamma}$, we show that $G$ can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's $\vert G\vert_{\Theta}$ is the finest such compactification, and our $\vert G\vert_{\Gamma}$ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.