Number of connected components of the complement of a closed curve.

Question

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 \backslash\ \gamma$, where $\gamma$ just denotes the curve's image. I get the feeling that the number of connected components should be finite but I've been unable to proof it.

Answer 1

I don't know algebraic topology so you be the judge of this. Isn't it possible to define $\gamma$ such that between $[1/2,1]$, I have a circle of radius $1$ centred at $(1,0)$, between $[1/3,1/2]$, I have a circle of radius $1/4$ centred at $(1/2^2,0)$,...between $[1/(n+1),1/n]$, I have a circle of radius $1/n^2$ centred at $(1/n^2,0)$.

As you see we are at origin for reciprocals of integers, and we have infinite components.