Is $E = \{ (x,y) \in \mathbb{R}^2 \mid xy \leq1 \}$ connected? I'm trying to solve this point set topology problem.
I need to prove whether or not $E_1 = \{ (x,y) \in \mathbb{R}^2 \mid xy \leq 1 \}$ is connected. We are using the euclidean topology, as usual. I feel like this set is connected, hence I tried to prove it by contradiction, but I struggled to find one. 
Moreover, I need to find the connected components of $E_2 = \{(x,y) \in \mathbb{R}^2 \mid xy \geq 1\}$. Here, I think that $E_2$ is not connected itself, so I cannot find its connected components. Am I right?
 A: Hint 1. Note that if $(x,y)\in E_1$ then $(tx,ty)\in E_1$ for all $t\in [0,1]$, that is the segment with endpoints $(x,y)$ and $(0,0)$ is in $E_1$.
Hence $E_1$ is a star domain.
Hint 2. $E_2$ is not connected: show that $(1,1)\in E_2$ and $(-1,-1)\in E_2$ are in different  connected components.
A: There are three obvious ways to prove a set is connected:


*

*Write the set as the continuous image of a connected set and apply the intermediate value theorem.

*Show that the set is path connected. You can do this by:
i. showing the set is convex, or
ii. finding a path between any two points in the set, or
iii. finding a path between any point and some fixed point (such as the origin).

*Try to apply the definition.
In this case, method 2.iii works.
For the second question, note that connected component means a maximal connected subset, not that the original set is connected. In this case, $E_2$ has two connected components:
$$ \{(x,y) : x \ge 0, xy \ge 1\} \text{ and } \{(x,y) : x \le 0, xy \ge 1\}. $$
Hopefully the picture makes this clear.
