In math, a curve in a space X is a mapping from an interval to the space X.
One -to be more acute- must add that the map gotta be injective and a derivative non-zero
For example if and
is a parabola in the 2D euclidean space.
A curve -whether it lives- is as slim as the real line at least
locally. That is why a mathematician says that a curve is localy
homeomorphic to a 1D euclidean space.