Determine whether the set $X=\{(a,b) : |b|>e^a \}\subset \mathbb R^2$ is connected

Question

Determine whether the set (as a subspace of $\mathbb R^2$) is connected.

$$X=\{(a,b) : |b|>e^a \}$$

Thoughts: Not sure how to go about this question. I suppose look for a partition. Anyone got any ideas?

Answer 1

Partial answer: The region $X$ is the set of points that are above the curve of equation $y=e^x$, or below the curve of equation $y=-e^x$. Hence $X$ can be written as $\{(a,b)\mid b>0, b>e^a\}\cup\{(a,b)\mid b<0, b<-e^a\}$, the union being disjoint. If you can show that these two sets are open, $X$ is not connected and you are done.

Final hint: $\{(a,b)\mid b>0, b>e^a\} = \{(a,b)\mid b>0\}\cap X$.

Answer 2

Hint:

$$X=\left\{(a,b)\;;\;b<-e^a\right\}\cup\left\{(a,b)\;;\;b>e^a\right\}$$

Answer 3

Just as an alternative, consider the following approach. In $\Bbb R^2$ the set is connected iff it is path connected. We know that points $(0,2)$ and $(0,-2)$ are in $X$, so that there must exist a point $(a,0)\in X$, which is clearly impossible by definition of $X.