TITLE: half space is not homeomorphic to euclidean space
QUESTION [8 upvotes]: We define, half space $H^n$ = $\{(x_1,x_2,...,x_n) | x_n \geq 0\}$. Can anyone suggest, how to prove that $H^n$ is not homeomorphic to the euclidean space $R^n$.

REPLY [3 votes]: If there exists an homeomorphism from $ H^n $ to $ R^n $, then removing $ \{x:x_n=0\} $ from the domain you get a homeomorphism from a connected space to a disconnected one. This follows from Jordan Brouwer separation theorem: if  a closed set of $ R^n $ is homeomorphic to $ R^{n−1} $ then it disconnects $ R^n $ into two connected components.