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?
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?
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.]