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Prove the following homeomorphism: $ R_{ \geqslant 0} \times R_{ \geqslant 0} \, \cong \,R_{ \geqslant 0} \times R $

where R are the real numbers and $ R_{ \geqslant 0} = \left\{ {x \in R:x \geqslant 0} \right\} $

If $ \,R_{ \geqslant 0} \, \cong \,\,R $ it´s done , but I think that this is not true, since it´s obvious that a continuous bijective function $ f:A \subset R \to R $ must be strictly increasing or strictly decreasing, so considering $ f(0) $ it´s easy to see that a continuous injective function between this two sets cannot be surjective, so there are not homeomorphic, but does not imply necesarly that the product it´s not. What can I do?

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    Here's a very similar post: http://math.stackexchange.com/questions/115577/example-of-diffeomorphism You'll need to modify the solutions offered in the comments slightly, but most of the maps suggested can be extended to the boundary.2012-03-03

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