Question

is (a,b) open on R^2

Answer 1

yes

Answer 2

Suppose that \(a<b\) where \(a,b \in \mathbb{R}\).
Choose any \(x\in (a,b)\) and consider \(\epsilon = \min(|x-a|,|x-b|)=\min(x-a,b-x)\).
Then it is obvious that \(B(x,\epsilon)\subset (a,b).\)

Answer 3

\((a,b)\subset \mathbb{R}\) not \(\mathbb{R}^2\)