||In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1 < α ≤ ∞. Related to the 2 latter, we improve a well-known result of A. Cohen by showing ∞ that the correspondence m → ϕ = 1m(2−j ·), between lowpass ﬁlters in H α (T) and Fourier transforms of α-localized scaling functions (in H α (R)), is actually a homeomorphism of topological spaces. We also show that the space of such ﬁlters can be regarded as a connected inﬁnite dimensional manifold, extending a theorem of A. Bonami, S. Durand and G. Weiss, in which only the case α = ∞ is treated. These two properties, together with a careful study of the “phases” that give rise to a wavelet from the MRA, will allow us to prove that the space Wα , of α-localized wavelets, is arcwise connected with the topology of L2 ((1 + $2 )α dx) (modulo homotopy classes). This last result is new even for the case α = ∞, as well as the considerations about the “homotopy degree” of a wavelet.