simple and closed curves

Question

I am struggling in understanding how to distinguish between simple/non simple and closed curves. For example, is a circle a simple curve? It does intersect itself in the starting point

thanks

Answer 1

The distinction is that one needs to pay attention to the domain of the curve, a point that is often glossed over.

A curve $p: I \to X$ is a continuous function from the interval $I \subset \mathbb{R}$ into the topological space X (where it makes sense to talk about continuity). The curve $p$ is simple if and only if $p$ is injective (the word cross is ambiguous). The curve $p$ is closed if $I = [a,b]$ such that $p(a)=p(b)$. Note that intuitively a simple, closed curve would require that two different points $x,y \in I$ have distinct images $p(x)$ and $p(y)$ (the curve doesn't cross itself at any point) and the image of the endpoints of the interval are the same.

A simple closed curve is a closed curve that is also injective on the domain $[0,1)$ (note the end point $1$ is missing!).

So, for example, if I take $p:[0,1] \to \mathbb{R}^2$ defined by $p(t) = (\cos (2 \pi t), \sin (2 \pi t))$, then $p$ is not simple, it is closed and it is a simple closed curve. Hence, it is a simple closed curve.

Take care to distinguish the range from the curve itself. So, in the above examples, the ranges are all the circle.

Legal disclaimer:

There are lots of definitions of curves. For example, in Munkres we have a simple closed curve as a space that is homeomorphic to the unit circle $S^1$. The term arc is used for a space homeomorphic to $[0,1]$ and a curve is often a term used for a $1$-manifold.

However, the above definitions will suffice for many useful cases.