  Working in a variant of the intersection type assignment system of Coppo,
Dezani-Ciancaglini and Venneri [1981], we prove several facts about sets of
terms having a given intersection type. Our main result is that every strongly
normalizing term M admits a *uniqueness typing*, which is a pair $(\Gamma,A)$
such that
  1) $\Gamma \vdash M : A$
  2) $\Gamma \vdash N : A \Longrightarrow M =_{\beta\eta} N$
  We also discuss several presentations of intersection type algebras, and the
corresponding choices of type assignment rules.
  Moreover, we show that the set of closed terms with a given type is uniformly
separable, and, if infinite, forms an adequate numeral system. The proof of
this fact uses an internal version of the B\"ohm-out technique, adapted to
terms of a given intersection type.
