Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?

I am interested in particular in the case when $X$ and $Y$ are both complete metric spaces.

Any help will be very much appreciated!

EDIT: Just to be clear, by proper map I mean a map such that the preimage of a compact set is a compact set.

  • 2
    $\begingroup$ If for whatever reason your homotopy $F$ also happens to be a closed map this is true, but as below there is no reason to believe that a map $A \to I$ with compact fibers must have total space compact. $\endgroup$ – Mike Miller Jan 21 at 22:51

The following counterexample is stolen from page 1 of this paper by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ contains the sequence $(1-\frac{1}{n}, -n)_{n\in\mathbb N}$. Of course, each parabola $x\to (1-t)x^2+x$ is proper.

  • $\begingroup$ Nice! Thanks a lot! $\endgroup$ – Onil90 Jan 22 at 7:31

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