Which of these subsets $M \subset \mathbb{R}^2$ is a submanifold?
a. $M = \{(x,y)\,|\, x^2+y^2=1\}$
b. $M = \{(x,y)\,|\, x=0 \mbox { oder } y= 0\}$
c. $M = \mathbb {Q}^2$
I'm kind of sure that a is a submanifold and that c is not. I'm not sure about b but I think that it is a submanifold. Is this correct?
No. (b) is not: consider the origin $(0,0)$.
If you take a neighborhood of the origin, it is not locally an open interval of $\mathbb{R}$.