Suppose $f\colon X \rightarrow Y$ is a continuous map of topological spaces and $s\colon Y \rightarrow X$ is a continuous section to $f$, i.e., $f\circ s = 1$. If $f$ is proper does this mean that $s$ is proper as well? (A continuous map is proper if the preimage of any compact set is compact.)


This is true for schemes and I was wondering whether the same is true for topological spaces.



Is a section of a proper map proper?
Not in general. As Mariano stated in the comments, it's true if $X$ is $T\_2$. Here's a counterexample when $X$ is only $T\_1$:


Let $X$ be the space obtained by gluing two closed unit intervals (call them $I\_1,I\_2$) along some open subinterval. Let $Y$ be a closed unit interval, $f:X\rightarrow Y$ the obvious quotient map, and $s$ one of the two obvious lifts, say to $I\_1$. Since $X$ is compact and $Y$ is Hausdorff, $f$ is proper. However, the preimage of $I\_2$ in $Y$ will be an open interval, so $s$ is not proper.


