Curve

In math, a curve in a space X is a mapping from an interval to the space X. In symbols

$\mathrm {curve} :I\to X$

One -to be more acute- must add that the map gotta be injective and a derivative non-zero

For example if $I=\mathbb {R}$ and $\mathrm {curve} (t)=(t,t^{2})$ is a parabola in the 2D euclidean space.

A curve -whether it lives- is as slim as the real line $\mathbb {R}$ at least locally. That is why a mathematician says that a curve is localy homeomorphic to a 1D euclidean space.