Prove $\{(x,y): x>0\}$ is connected As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that $\{(x,y): x>0\}$ is connected. My tools for this are:

Definition 1: Two disjoint sets $A$ and $B$, neither empty, are said to be mutually separated if neither contains a boundary point of the other.
Definition 2: A set is disconnected if it is the union of separated subsets.
Definition 3: A set is connected if it is not disconnected.

Because Definition 3 is a negation, I'm encouraged to do this by contradiction. Suppose $\{(x,y): x>0\}$ is disconnected. Then it is the union of mutually separated sets. I don't know where to go from here or if there is a way to show directly that the set can't be expressed as the union of mutually separated sets directly. Any guidance would be appreciated.
 A: This may not be exactly what you are looking for, but here's one way:
The function $f: \mathbb{R}^2\to\mathbb{R}^2$ given by $f(x, y)=(e^x, y)$ is continuous. The continuous image of a connected set is connected. The image of $f$ is precisely $\{(x, y)\in\mathbb{R}^2: x>0\}$. 
A: The weird thing about connectedness is that proof by contradiction is sort of the direct way to prove it. The definition of connected is negative. A set is connected if it is not satisfying a certain condition. So to contradict that would be to assume a set is satisfying a certain condition, and therein is why it gets a pseudo-direct feel to it. I recommend doing the proof by contradiction as you were already encouraged.
The first thing I'd do is draw in the $x$-$y$ plane (or have a clear mental picture of) your set. You need to prove that no matter how you split that set into two pieces that those two pieces cannot both be mutually separated. So proceed as you say by assuming the existence of two mutually separated sets $A,B$ whose union is your set. You didn't provide your definition of boundary point, but use the fact that $A$ and $B$ cannot contain a boundary point of the other. Then choose a boundary point and see what else you can say about it. Keep trying to find things you can say until you find you have reached a contradiction. The contradiction may not be immediately clear, but if you play around with these definitions for a while something should pop out for you.
