Prove that $A=\{(x,y)\in\mathbb{R}:(y\neq 0)\vee (x>0)\}$ is connected. I'm working on a multivariable integration problem and, at some point, I need to use Poincaré Lemma. To justify that I verify the Lemma's hypothesis conditions, I need to prove that the set
$$A=\{(x,y)\in\mathbb{R}:(y\neq 0)\vee (x>0)\}$$
is connected. In fact, proving it's simply connected will work as well for my excercise, I don't know wich way is easier, and in any case how to prove it cannot be represented as the union of two or more disjoint non-empty open subsets. I will thank any help or advice.

The space you presented is just $\mathbb{R}^2\setminus \{ (x,0) : x\leq0 \}$. That is you are removing from the real plane the half line starting at the origin and going through the negative part of the $x$ axis. 
You can easily show it is path connected, thus it is connected.
It is also simply connected but from the phrasing of your questions I would think you are not entirely familiar with this concept, am I right?
Simply connected means that the first homotopy group of the space is trivial. If you do not know the definition of this group: this is equivalent to the fact that every loop in your space can be contracted to its basepoint.
