TITLE: How to prove figure eight is not a manifold?
QUESTION [9 upvotes]: Possible Duplicate:
A wedge sum of circles without the gluing point is not path connected 

I know that figure eight is not a manifold because its center has no neighborhood homeomorphic to $\mathbb{R}^n$. But how to prove this strictly?

REPLY [12 votes]: Suppose that there was a neighborhood $U$ of the center point $P$ that was homeomorphic to $\mathbb{R}^n$.  Consider $U \setminus \{P\}$.  How many connected components does it have?  How many connected components are there in $\mathbb{R}^n \setminus \{\text{point}\}$?  [Be careful to note that the answer is different for $n=1$ than for $n > 1$, but that doesn't ultimately cause any trouble.]