I have a basic topology question in the setting of topological group actions.
Let $G$ be a topological group acting properly on a topological space $X$. Let $K$ be a maximal compact subgroup of $G$. Suppose you have a $G$-equivariant map $f:X\rightarrow G/K$, where $G$ acts in the natural way on $G/K$. Denote by $\pi:X\rightarrow Y =: X/G$ the quotient map, and write $S:=f^{-1}(eK)$.
Question: Why is the map $\pi|_S:S\rightarrow Y$ a closed map?
(If needed, we are allowed to use that $G$ is locally compact and $X$ is completely regular Hausdorff.)