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Possible Duplicate:
Decomposition of a manifold

For topological spaces $X,Y$, if their product space $X \times Y$ is a manifold, is it necessarily that $X,Y$ are manifolds?

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marked as duplicate by t.b., Nate Eldredge, Jonas Teuwen, Henning Makholm, Davide Giraudo Aug 2 '12 at 22:32

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
Also related: math.stackexchange.com/q/169195/5363 – t.b. Aug 2 '12 at 22:26

No. The dogbone space $D$ is a topological space that is not a manifold but $D \times \mathbb{R} \cong \mathbb{R}^4$.

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