I want to obtain contradiction... Let $X \subseteq \mathbb{R}^n$. I managed to show on one hand $X$ is an $k$ dimensional manifold, and then also that $X$ is an $m$ dimensional manifold, where $m \not = k$. Now I want to say this is not possible hence we have obtained contradiction.
Although I'm very sure this is the case, I wasn't sure how to show this (that $k$ dimensional manifold can not be $m$ dimensional manifold)... Any comments on how to see this are appreciated. Thank you.