Show that $A=\{(x,y) \in \mathbb{R}^2|x=1/2, 0\leq y<1\}$ is Borel

Question

I want to show that $A=\{(x,y) \in \mathbb{R}^2|x=1/2, 0\leq y<1\}$ is Borel using intersections

I have gotten: $$A=\bigcap_{n \in \mathbb{N}} [1/2,1/2+1/n) \times [0,1)$$

However I have some doubts if this actually creates an intersection on the y-axis because it is static

Any help/hint would be appreciated

Answer 1

$$(x,y)\in A=\bigcap_{n \in \mathbb{N}} \bigg(\bigg[\frac{1}{2},\frac{1}{2}+\frac{1}{n}\bigg) \times [0,1)\bigg)=\bigg(\bigcap_{n \in \mathbb{N}} \bigg[\frac{1}{2},\frac{1}{2}+\frac{1}{n}\bigg)\bigg) \times [0,1)$$$$\iff x\in \bigcap_{n \in \mathbb{N}} \bigg[\frac{1}{2},\frac{1}{2}+\frac{1}{n}\bigg)\ \text{and}\ y\in [0,1)$$

$$\iff x=\frac{1}{2}\ \text{and}\ y\in [0,1).$$

So you are right.