# Proper homotopy

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.

• 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.
– mme
Jan 21, 2019 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.