$M:=\{(x,y)\in \Bbb R^2: x\cdot y=0\}$ is not a submanifold

Question

I observe the following set: $$M:=\{(x,y)\in \Bbb R^2: x\cdot y=0\}$$ $M$ is nothing else than the intersection of the $x$ and the $y$ axis.The question is, can this set be a submanifold of $\Bbb R^2$? I guess no for the point in the origine i.e. but I can't help me to give some better explanation. As hint I could look at the tangent space but I still don't get.

Answer 1

Firs of all, that set is not the intersection of the $x$-axis and the $y$-axis; its their union.

If $V$ is a neighborhood of $(0,0)$ on $M$, then $V\setminus\{(0,0)\}$ has, at least, four connected components. If $M$ was a manifold, then it would either be $1$-dimensional, in which case $(0,0)$ would have a neighborhood such that, if you remove $(0,0)$ from it, you get two connected components, or it would be $n$ -dimensional with $n>1$, in which case $(0,0)$ has a neighborhood such that, if you remove $(0,0)$ from it, you still get a single connected component.