The rescaled h-increments Y_t,h(u) = (2h log(1/h))-^1/2{W(t + hu) - W(t)}, for u ∈ [0,1], of a Wiener process {W(t) : t ≥ 0}, are considered as elements of the space C₀[0,1] of all continuous functions g on [0,1] with g(0) = 0. We endow C₀[0,1] with the topology defined by a norm ∥·∥_ν chosen within a general class C for which the limit law limh↓0{sup₀≤t≤1 ∥Y_t,h∥_ν} < ∞ holds with probability 1. We show that, for each f ∈ C₀[0,1] with ∫₀{d/du f(u)}²du≤ 1, the set ℒ_ν(f)={t ∈ [0,1]: lim inf_h↓0∥Y_t,h-f∥_ν=0} contains, with probability 1 for each ν ∈ C, a subset ℒ(f), independent of ∥·∥ C and with Hausdorff dimension equal to dim(ℒ(f)) = 1 - ∫₀ ¹ {d/du f (u)}²du.