  The directions of an infinite graph $G$ are a tangle-like description of its
ends: they are choice functions that choose compatibly for all finite vertex
sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is
induced by a ray, there exist directions of graphs that are not uniquely
determined by any countable subset of their choices. We characterise these
directions and their countably determined counterparts in terms of star-like
substructures or rays of the graph. Curiously, there exist graphs whose
directions are all countably determined but which cannot be distinguished all
at once by countably many choices. We structurally characterise the graphs
whose directions can be distinguished all at once by countably many choices,
and we structurally characterise the graphs which admit no such countably many
choices. Our characterisations are phrased in terms of normal trees and
tree-decompositions. Our four (sub)structural characterisations imply
combinatorial characterisations of the four classes of infinite graphs that are
defined by the first and second axiom of countability applied to their end
spaces: the two classes of graphs whose end spaces are first countable or
second countable, respectively, and the complements of these two classes.
