It is proved that associated with every wavelet set is a closely related "regularized" wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L2() of MSF wavelets for which the Fourier transform has support contained in E F. Our technique applies, in particular, to the Shannon and Journe wavelet sets.
