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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?

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marked as duplicate by Brian M. Scott, Emily, Jason DeVito, Henry T. Horton, rschwieb Oct 22 '12 at 16:18

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up vote 9 down vote accepted

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.]

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I got it, but can I assume that $U$ is a small nbd of $P$? Then the connected components will be 4, but if $U$ is strangely shaped, the components can be more than 4. – Gobi Oct 22 '12 at 12:55
@Gobi: $P$ has arbitrarily small $+$-shaped nbhds; no point of $\Bbb R^n$ has such a nbhd. See this answer and the comments following it. – Brian M. Scott Oct 22 '12 at 12:58

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