In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb R$ is homeomorphic to $\Bbb R^4$.

Is there a space $X$, not homeomorphic to $\Bbb R$, such that $X\times \Bbb R $ is homeomorphic to $\Bbb R^2$?

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    No, such spaces do not exist. You can prove it yourself by thinking about Hanno's answer here. – Moishe Cohen Aug 25 '15 at 20:49

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