  A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a
set of vertices such that any two of its subsets of the same size $\ell \leq k$
can be connected by $\ell$ disjoint paths in the whole graph.
  We characterise the existence of $k$-connected sets of arbitrary but fixed
infinite cardinality via the existence of certain minors and topological
minors. We also prove a duality theorem for the existence of such $k$-connected
sets: if a graph contains no such $k$-connected set, then it has a
tree-decomposition which, whenever it exists, precludes the existence of such a
$k$-connected set.
