We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-5, author = {Jan van Mill}, title = {On Countable Dense and Strong Local Homogeneity}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {53}, year = {2005}, pages = {401-408}, zbl = {1108.54024}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-5} }
Jan van Mill. On Countable Dense and Strong Local Homogeneity. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) pp. 401-408. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-4-5/