  A phantom map is a potentially nontrivial map which induces the zero map on
every homology theory and on homotopy groups. Zabrodsky has shown that in the
presence of particular finiteness conditions on spaces $X$ and $Y$ every map
$X\to Y$ is a phantom map. More specifically, Zabrodsky essentially requires
$Y$ to be a finite CW complex and $X$ to be a Postnikov space. We show
Zabrodsky's observations hold under less restrictive finiteness conditions on
the spaces $X$ and $Y$, making use of the Zabrodsky lemma and the machinery of
resolving classes.
  As an application we identify, up to extension, the group of self-homotopy
equivalences of spaces belonging to a particular family.
