I have to prove that $(\mathbb{R}\times(-\infty,0)) \; \cup\{(x,y)\in\mathbb{S}^{1}: x\geq0\}\subseteq\mathbb{R}^2 $ is contractible.
What I thought is that the first element of the union is contractible (because there exists one proposition that states that if we have two spaces, $X$, $Y$ for example, and one of them is contractible, then for any $Y$, $X\times Y$ has the same homotopy type of $Y$, so what I said is true, because both are contractible).
My problem is to prove that the second part of the union is contractible.
My idea is, perhaps I would have started by carrying the points of the positive semi-circle to the interval $(0,1)$ and the negative semi-circle to de points of the form $(0,y)$ with $y\in[0,1]$. Then what I have, is the first element of the union and then I have finished; nevertheless, really, I don't know if I'm doing well, because I would have proving two homopies of which I'm not sure, and really, I don't know of to proceed (I have an intuition, project the points that I said where I said).
Sorry, if I'm lying in something of my statements.