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I'm in a multivariable calculus course and I quite struggle with proofs where to show that something is not a sub-manifold. As you can't use the pre-image theorem, can someone please explain a relative solid "method" to start such a proof. For example for this sub-manifold M ={(x,y)∈ R^2 |xy=0}

Thank you very much in advance.

user406823
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If we consider the set $M = \{(x,y): xy = 0\}$ then we observe that it is just the union of the sets $\{(x,0)\} \cup \{(0,y)\}$ i.e we can embed $M$ so that $M \subset \mathbb{R}^3$ is the union of the coordinate lines $(0,t,0)$ and $(s,0,0)$.

Observe that $M$ cannot be a $1$-manifold since it is non-compact and would have to homeomorphic to $\mathbb{R}$ which we see is not true. If you think $M$ is a surface, take an open ball $B$ about $(0,0)$ and consider $B \cap M$. No matter how small the ball is, this intersection looks like an $\textbf{X}$ which is not homeomorphic to a disk.

Faraad Armwood
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    This is not a proof of anything. And why did you put things in $\Bbb R^3$? – Ted Shifrin Apr 29 '17 at 22:42
  • What do you mean, this isn't the proof of anything? This proves the statement, anything else if just fancy. A question of rigor maybe, but to say it proves nothing, I don't agree. Also, I put things in $\mathbb{R}^3$ since the OP said sub-manifold i.e subset of some other manifold. The canonical choice in this case is $\mathbb{R}^3$. – Faraad Armwood Apr 30 '17 at 00:01