[cka]

A curve is a 1-dimensional space. Even though you draw a picture of it in a 2 (or maybe higher?) dimensional space, you should think of the set of points in the curve as being the only points that we care about.

In a closed curve, you can never fall out of the space by moving around in it. In a curve with endpoints, you can fall out of the space by walking across one of the endpoints, so the endpoints are considered to be boundary components.