Is the following space connected

Question

Is the space $(\mathbb Q\times\mathbb Q)\cup$$(\mathbb R- \mathbb Q\times \mathbb R-\mathbb Q)$ connected?

I am really stuck at it. Any hints?

Answer 1

Let $A:=(\mathbb{Q} \times \mathbb{Q}) \cup(\mathbb{Q}^C \times \mathbb{Q}^C)$ be the space we're interested in. We can show that $A$ is path-connected.

Let $(a,b), (c,d)\in A$. We want to find a continuous map $\varphi:[0,1]\to A$ satisfying $\varphi(0)=(a,b)$ and $\varphi(1)=(c,d)$.

For this, connect the points $(a,b), \left(a+\frac{d-b}{2},b+\frac{d-b}{2}\right),(a,d),\left(a+\frac{c-a}{2},d+\frac{c-a}{2}\right),(c,d)$ via straight line sigments (in the order written down). Let $L$ be the union of those line segments. Choose $\varphi$ moving through $L$ from $(a,b)$ to $(c,d)$.

It remains to show that $L\subseteq A$. Can you prove this? Note that the line segments in $L$ are all parallel to the lines given by $y=x$ and $y=-x$.