Abstract In this paper, we obtain that the space $\Cal{W}$
of orthonormal wavelets enjoys the complete invariance property
with respect to homeomorphisms. Further, it is obtained that the
cylinder, the cone and the suspension of $\Cal{W}$ possess the
complete invariance property. Certain subspaces of $\Cal{W}$ are
also considered in this connection. ![Creative Commons License](https://i.creativecommons.org/l/by/4.0/80x15.png)
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