There is a set U of reals such that for every analytic set A there is a continuous function f which maps U bijectively to A.
@article{bwmeta1.element.bwnjournal-article-fmv159i2p153bwm, author = {Theodore Slaman}, title = {On a question of Sierpi\'nski}, journal = {Fundamenta Mathematicae}, volume = {159}, year = {1999}, pages = {153-159}, zbl = {0926.03059}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv159i2p153bwm} }
Slaman, Theodore. On a question of Sierpiński. Fundamenta Mathematicae, Tome 159 (1999) pp. 153-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv159i2p153bwm/
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