# Proper maps for connectedness

A map $f : X \to Y$ between topological spaces $X$ and $Y$ is said to be proper if the inverse image of a compact subset of $Y$ is compact in $X$.

Does there exists an analogous concept for connectedness?

A map $f : X \to Y$ between topological spaces $X$ and $Y$ is said to be ???? if the inverse image of a connected subset of $Y$ is connected in $X$.

If such a concept does not exist or hasn't been studied much, is there a reason for this? Why in that case has compactness seemingly been deemed of more important than connectedness?

• I would think it's a very rare property. Something as innocuous as $f(x)=x^2$ on the reals fails it: $f^{-1}(1,+\infty)$ is disconnected. – Randall Sep 22 '17 at 11:56
• Yes, there is such a concept; it is useful in algebraic topology, start by reading here. – Moishe Kohan Sep 23 '17 at 5:29

Sometimes a map between topological spaces $f\colon X \to Y$ is called monotone if it has connected fibers, i.e. $f^{-1}(y)$ is connected for every $y ∈ Y$. (Similarly, $f$ is called perfect if it has compact fibers and it is closed and continuous. Such maps are proper.)
If $f$ is a monotone quotient map and $Y$ is connected, then $X$ is connected as well. Hence, if $f$ is monotone and hereditarily quotient, then the preimage of a connected set is connected. (Hereditarily quotient means that every corestriction $f\colon f^{-1}[B] \to B$ is quotient. Open or closed continuous maps are such.)
• @Arrow: Sure, that's a matter of taste. But a monotone map does not have to be onto, e.g. $\operatorname{arctan}\colon ℝ \to ℝ$ is monotone. – user87690 Dec 13 '18 at 22:27
• Dear @user87690, on a very related manner, could you please take a look at Engelking 6.2.23? The mapping $h(z)=g_2g_1^{-1}(z)$ seems to be undefined if $g_1^{-1}(z)$ is empty, and this seems to invalidate the proof. So it appears having (nonempty) connected fibers is crucial for essential uniqueness of monotone-light factorizations. What do you think? – Arrow Dec 14 '18 at 11:17