TITLE: Can a mathematical definition be wrong?
QUESTION [94 upvotes]: This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently published a paper proving that quantum Turing machines could compute uncomputable functions. In subsequent papers the definition of quantum Turing machine was revised to include the uniformity condition, correcting what was clearly a mathematical error the original authors made.
It seems to me that in the idealized prescription for doing mathematics, the original definition would have been set permanently, and subsequent papers would have needed to use a different term (say uniform quantum Turing machine) for the class of objects under study. I can think of a number of cases where this has happened; even in cases where, in retrospect, the original definition should have been different.
My question is: are there other cases where a definition has been revised after it was realized that the first formulation was "wrong"?

REPLY [2 votes]: For me the more interesting question is what makes definitions "right" — like how the definition of e.g. function, manifold, or Lie group are definitions that are widely believed to be very unlikely to undergo any future revisions.*
But maybe I'm being fussy. When a mathematician learns a mature definition for a subtle concept, there is a feeling as if something is clicking into place that can be very satisfying.

As for manifolds, there are actually a number of definitions. Most obviously, depending on the pseudogroup of transition functions allowed for its atlas, most commonly continuous, merely homeomorphic to a simplicial complex, piecewise linear, differentiable of class Ck with 1 <= k <= infinity, real analytic, or complex analytic. And there is the question of whether infinite dimensions should be included. But also, there are diverging opinions on whether a manifold must be second countable, and in some quarters even on whether a manifold must be Hausdorff. Personally I favor removing all such restrictions but the requirement that it be locally Euclidean of some nonnegative dimension. But this can be terminologically inconvenient.