How to extend an interval to a circle in $\mathbb{R}^2$

Let $$\gamma : [0,1] \rightarrow \mathbb{R}^2$$ be a $C^0$ imbedding. How can I show that there exists another imbedding $$\eta : [0,1] \rightarrow \mathbb{R}^2$$ with $\eta ((0,1)) \subset \mathbb{R}^2 - \gamma [0,1]$ and $\gamma (0)= \eta (1), \gamma (1) = \eta (0)$? Thank you for yor help!

• What kind of machinery are you allowed to use? There is a pretty easy way to do this but it may not be accessible to you. – Cameron Williams Jun 2 '13 at 16:04
• May be relevant: math.stackexchange.com/questions/287062/… – Seirios Jun 2 '13 at 16:06
• hmm. maybe give me a sketch and I'll see if it sounds like something I might understand? – Famous Mortimer Jun 2 '13 at 16:06
• @Cameron Williams : Actually I think I'd almost prefer a short high tech proof. I think those tend to be more enlightening – Famous Mortimer Jun 2 '13 at 16:48