Proving path-connectedness of $\mathbb{R}^2\setminus\mathbb{E}$ where $\mathbb{E}$ is the set of points with both coordinates rational 
Let $\mathbb{E}$ be the set of all points in $\mathbb{R}^2$ having both coordinates rational. Prove that the space $\mathbb{R}^2\setminus\mathbb{E}$ is path-connected. 

I think I just need to find a continuous function $f$ from $[0,1]$ into $\mathbb{R}^2\setminus\mathbb{E}$ with $f(0) = a$ and $f(1) = b$. This part is probably wrong as I am dealing with two dimension.
I would be really appreciate hints.
Thanks.
 A: (Abstract) Hint: Given two points on this grid, how would you construct a path along the grid between them?

A: Let $a=(x_1,y_1)$ and $b=(x_2,y_2)$ be two points of $\Bbb{R}^2\setminus\Bbb{E}$. By definition of $\Bbb{E}$, at least one coordinate of both $a$ and $b$ must be irrational, so suppose for instance that $x_1$ and $y_2$ are irrational.
You are right that we need to find a continuous function $f\colon [0,1]\to \Bbb{R}^2\setminus\Bbb{E}$, such that $f(0) = a$ and $f(1) = b$. You can compose this function as follows:
First, move along a straight, vertical line segment from $(x_1,y_1)$ to $(x_1,y_2)$. This line segment will be contained in your space, because $x_1$ is irrational.
Next, move along the straight, horizontal line segment from $(x_1,y_2)$ to $(x_2,y_2)$. The function $f$ will be the composition of these two line segments. This is the idea, but you should define $f$ formally.
You can also use this idea for the other cases of which coordinates are irrational.
A: Path-connectedness, by definition, means for every pair of points $a, b$ in $\Bbb R^2 \setminus E$, there is a continuous map $f : [0, 1] \to \Bbb R^2 \setminus E$ with $f(0) = a$, $f(1) = b$. So what you said is not wrong.
To prove that such paths exist, note that there are uncountably many lines passing through $a$ in $\Bbb R^2$. As $E$ is countable, there are uncounably many lines through $a$ not intersecting $E$. Similarly, there are uncountably many lines passing through $b$ which does not hit $E$. You can pick two such lines passing through $a$ and $b$ respectively so that they are not parallel, i.e., so that they intersect.
This gives you a path between $a, b$ in $\Bbb R^2$ not hitting $E$, as desired.
A: That's actually correct. For any $a,b\in\Bbb{R}^2 - \Bbb{E}$, you need to find such a path. My hint to finding such a path is, try to think about what the space $\Bbb{R}^2 - \Bbb{E}$, the points with at least one irrational coordinate, might look like. Then see if you can visualize how you might go from one point with at least one irrational coordinate to another point with at least one irrational coordinate so that you always have at least one irrational coordinate.
