Proceedings of the American Mathematical Society

(Communicated by J. Marshall Ash) Abstract. A construction of wavelet sets containing certain subsets of R is given. The construction is then modified to yield a continuous dependence on the underlying subset, which is used to prove the path-connectedness of the s-elementary wavelets. A generalization to R n is also considered. A function f ∈ L2 (R) isadyadic orthogonal wavelet (or simply a wavelet if no confusion can arise) if {2n/2f(2nx + l)}l,n∈Z is an orthonormal basis for L2 (R). Alternatively, if we define D: L2 (R) → L2 (R) byD(f)(t) = √ 2f(2t) andT: L2 (R)→L2(R)byT(f)(t) =f(t−1), then by definition f is a wavelet if and