  The fractal necklaces in R^d (d>1) introduced in this paper are a class of
connected fractal sets generated by the so-called necklace IFSs, for which a
lot of basic topology questions are interesting. We give two subclasses of
fractal necklaces and prove that every necklace in these two classes has no cut
points. Also, we prove that every stable self-similar necklace in R^2 has no
cut points, whilst an analog for self-affine necklaces is false.
