In
mathematics, the statement that "Property
P characterizes object
X" means, not simply that
X has
property P, but that
X is the
only thing that has property
P. It is also common to find statements such as "Property
Q characterises
Y up to isomorphism". The first type of statement says in different words that the
extension of
P is a
singleton set. The second says that the extension of
Q is a single
equivalence class (for isomorphism, in the given example — depending on how
up to is being used, some other
equivalence relation might be involved).