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Let $U\subset \mathbb{R}^n$ (open set) , $V\subset \mathbb{R}^n$ and $f:U\longrightarrow V$ a homeomorphism then we can say that $V$ is a open set in $\mathbb{R}^n$ ?

Any hints would be appreciated.

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    Google for the theorem of «Invariance of Domain» and its consequences.2012-09-14

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Yes, this result goes by the name Invariance of Domain and is due to Brouwer. It's a non-trivial result.

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    Got it got it :)2012-09-14