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 $\R$ at least locally. That is why a mathematician says that a curve is localy homeomorphic to a 1D euclidean space.