I'm looking for a simple example of an open proper continuous application between topological spaces $\varphi:X\to Y$ such that :

• $Y$ is countractible and locally countractible ;
• for any $y\in Y$, $\varphi^{-1}(\{y\})$ is countractible ;
• $X$ is not countractible.

I have an example which is a little bit complicated but I wonder if there exists a simple one.

I'm looking for a simple example of an open proper continuous map between topological spaces $\varphi:X\to Y$ such that :

• $Y$ is contractible and locally contractible ;
• for any $y\in Y$, $\varphi^{-1}(\{y\})$ is contractible ;
• $X$ is not contractible.

I have an example which is a little bit complicated but I wonder if there exists a simple one.