As far as I know, there several definitions of a proper map.
- A function $f\colon X\to Y$ is proper if it is continuous and for any space $Z$, the product $f\times \operatorname{id_Z}\colon X\times Z\to Y\times Z$ is closed.
- A function $f\colon X\to Y$ is proper if it is continuous, closed, and preimages of points in $Y$ are compact.
- A function $f\colon X\to Y$ is proper if it is closed, continuous, and preimages of compact sets are compact.
- The same as 3., only not necessarily closed.
- The same as 4., only not necessarily continuous.
Bourbaki shows that the first two are equivalent (and that they are equivalent to two other definitions using filters). The last three are clearly progressively weaker.
Frankly, I'm quite confused now. So the question is: what are the implications between those properties (maybe under some additional assumptions about the spaces), and what other definitions of properness of a map may I come across in the future (and, obviously, how do those relate to the ones I have listed above). If scoured the internet for some time, but have not found any source which would clear up the confusion - I have only stumbled upon some partial results and vague remarks.