Q:

Retraction of $\{(z_1, ..., z_n) \in \Bbb C^n : \sum z_k^2 \neq 0 \} $ onto $\{(z_1, ..., z_n) \in \Bbb C^n : \sum z_k^2=1 \}$

Consider the space $X:=\{(z_1, ..., z_n) \in \Bbb C^n : \sum z_k^2 \neq 0 \} $ and its closed subspace $Y:=\{(z_1, ..., z_n) \in X : \sum z_k^2=1 \}$. Is there a retraction of $X$ onto $Y$?  Or, more generally, is there a continuous surjection $X \to Y$?  Intuitively it seems true, but I got stuck constructing such a map.  Any helps will be appreciated. ( Both $X$ and $Y$ have the Euclidean topology.) 

A:

No for $n=1$. We have $X = \{ z \in \mathbb C \mid z^2 \ne 0\} =\{ z \in \mathbb C \mid z \ne 0\} = \mathbb C \setminus  \{ 0 \}$ and $Y = \{ z \in X \mid z^2 =1 \} = \{-1,1\}$. Since $X$ is connected whereas $Y$ is not, there does not exist a continuous surjecton $X \to Y$.

