Simple connected space

I was wondering how to show that $\mathbb{C}\times\mathbb{R}^+$ is simple connected (every closed arc is continuously reducible to a dot).

The problem is more how can one write such paths in such space.

Can someone help ?

• A path in this space is described by a path in $\mathbb C$ and a path in $\mathbb R^+$. Their carthesian product is a path in $\mathbb C \times \mathbb R^+$... – AlexR Dec 4 '13 at 17:25
• Ok, I've got the point. Thank you ! – faero Dec 4 '13 at 17:33
• Happy to help. Is your question still open or should I post this as an answer? – AlexR Dec 4 '13 at 17:36
• I've found your comment complementary to Seirios' answer. – faero Dec 5 '13 at 20:01

Hint: $\mathbb{C} \times \mathbb{R}^+$ is star-convex. Therefore, it is sufficient to notice that any star-convex set is simply connected (see for example this question).