Chad, Knight & Suabedissen [Fund. Math. 203 (2009)] recently proved, assuming CH, that there is a 2-point set included in the union of countably many concentric circles. This result is obtained here without any additional set-theoretic hypotheses.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-6, author = {James H. Schmerl}, title = {Some 2-point sets}, journal = {Fundamenta Mathematicae}, volume = {209}, year = {2010}, pages = {87-91}, zbl = {1196.03057}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-6} }
James H. Schmerl. Some 2-point sets. Fundamenta Mathematicae, Tome 209 (2010) pp. 87-91. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-1-6/