Ends and tangles

We show that an arbitrary infinite graph can be compactified by its \({\aleph _0}\)-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its \({\aleph _0}\)-tangles. The \({\aleph _0}\)-tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The \({\aleph _0}\)-tangles that are ends are precisely the limits of principal ultrafilters.The \({\aleph _0}\)-tangles that correspond to a highly connected part, or \({\aleph _0}\)-block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations.