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Let $M$ be a connected $n$-manifold with a non-empty boundary. The double of it is given by $$ D(M) = M\,\,\,\cup_f\,\,\, M $$ where $f:\partial M\to\partial M$ is an identity map. I have to show that $D(M)$ is connected. Since it is a quotient space I had an idea to consider a connected space $X$ together with a quotient map $q:X\to D(M)$ but clearly $X = M\sqcup M$ is not a good candidate since it is disconnected. Could you help me to solve this problem?

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$D(M)$ is not connected unless $M$ is. Did you mean to assume $M$ is connected? – Chris Eagle Oct 25 '11 at 14:15
Assuming $M$ is connected, $D(M)$ is the union of two connected sets with nontrivial intersection. So by a standard lemma, it is connected. – Grumpy Parsnip Oct 25 '11 at 14:21
@ChrisEagle: thanks, $M$ should be connected. – Ilya Oct 25 '11 at 14:33
@JimConant: you're right. Could you please put this comment as an answer just expanding it a bit? – Ilya Oct 25 '11 at 14:35
up vote 1 down vote accepted

We are assuming $M$ is connected. The double $D(M)$ is equal to a union of two copies of $M$ that intersect in $\partial M\neq \emptyset$. On the other hand, it is a standard lemma in topology that the union of two connected sets that has a nontrivial intersection is connected. See for example Munkres's Topology (second edition) Theorem 23.3.

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Thank you, Jim. – Ilya Oct 26 '11 at 9:18

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