We construct a set B and homeomorphism f where f and have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-2-4, author = {Josh Campbell and David Swanson}, title = {Construction of an Uncountable Difference between $\Phi$(B) and $$\Phi$\_f(B)$ }, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {56}, year = {2008}, pages = {121-130}, zbl = {1151.28300}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-2-4} }
Josh Campbell; David Swanson. Construction of an Uncountable Difference between Φ(B) and $Φ_f(B)$ . Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) pp. 121-130. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-2-4/