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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.

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    $\begingroup$ What are your examples? (say the complicated one) $\endgroup$
    – wonderich
    Commented Jan 30, 2019 at 16:33
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    $\begingroup$ It is a Berkovich space over Banach ring which is not valued field. I'm currently working on this space. $\endgroup$
    – thibaud lemanissier
    Commented Jan 30, 2019 at 16:56
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    $\begingroup$ What about $X = \{(x,y) \in [0,1] \times [0,1] \;|\; x=0 \;\;\text{or}\;\; (x>0 \;\text{and}\; y=sin(1/x))\}$, $Y=[0,1]$ and $\varphi: X \to Y$ is the projection on the $x$-coordinate? All the conditions are clear except $\varphi$ being open, but I think it is open. $\endgroup$
    – Yonatan Harpaz
    Commented Jan 30, 2019 at 21:55
  • $\begingroup$ I may do a mistake but I think this map is not open $\endgroup$
    – thibaud lemanissier
    Commented Jan 30, 2019 at 22:01
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    $\begingroup$ This map is not open because the image of $U=X\cap \{(x,y)\in[0,1]\times[-1,1] |\ |y|<1/2\}$ is an open set but $\varphi(U)$ is not a neighborhood of $0$ $\endgroup$
    – thibaud lemanissier
    Commented Jan 30, 2019 at 22:20

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