  The paper deals with the program of determining the complexity of various
homeomorphism relations. The homeomorphism relation on compact Polish spaces is
known to be reducible to an orbit equivalence relation of a continuous Polish
group action (Kechris-Solecki). It is shown that this result extends to locally
compact Polish spaces, but does not hold for spaces in which local compactness
fails at only one point. In fact it fails for those subsets of $\mathbb{R}^3$
which are unions of an open set and a point. In the end a list of open problems
is given in this area of research.
