I'm doing problem 5-10 in Lee's Introduction to Smooth Manifolds: $\newcommand{\R} {\mathbb R}$ Let $M_a= \{(x,y) \in \R^2 | y^2 = x(x-1)(x-a) \} \subset \R^2$
For which values of a is $M_a$ an embedded submanifold of $\R^2$? For which values can $M_a$ be given a topology and smooth structure making it into an immersed submanifold? I've shown that for all $a \neq 0,1$, $M_a$ is an embedded submanifold and that $M_1$ can be made into an immersed submanifold.
This might be trivial, but I wanted to ask why $M_0 = \{(x,y) | x^2(x-1)=y^2\}$ cannot be made into an immersed submanifold of $\R^2$. It is clear from the graph that $(0,0)$ is an isolated point (in the subspace topology). However, how can I prove that there is no topology or smooth structure that can make $M_0$ into an immersed submanifold of $\R^2$? Whenever I try to prove this, I end up assuming something about the topology of $M_0$.