Let $G$ be a second countable locally compact Hausdorff group and $X$ be a second countable locally compact almost-Hausdorff $G$-space. If $X$ is totally disconnected and the orbit space X/G is Hausdorff, can we conclude that X/G is also totally disconnected?
We may assume that $G$ is totally disconnected.
If $X$ is Hausforff, then $X$ has a compact-open basis. It is not difficult to see that X/G is totally disconnected.
Recall that a topological space X is called almost-Hausdorff if every non-empty, closed subset $A$ of X contains a non-empty, relatively open Hausdorff subset.
See Definition 6.1, Theorem 6.2 and Lemma 6.3 of the following book about almost-Hausdorff. https://books.google.dk/books?id=0aBS-gYTWboC&pg=PA172&lpg=PA172&dq=almost-Hausdorff&source=bl&ots=e2VfXO0NYf&sig=pzVDwS5kpVMLO8gZ8QysuNi8n18&hl=en&sa=X&ei=heCfVIzkDMSeywOM-oCQCw&ved=0CDMQ6AEwAg#v=onepage&q=almost-Hausdorff&f=false