The main result of this paper consists of two theorems. One of them asserts that the functor U τ takes the 0-soft mappings between spaces of weight ≤ω 1 and Polish spaces to soft mappings. The other theorem, which is a corollary of the first one, asserts that the functor U τ takes the AE(0)-spaces of weight ≤ω 1 to AE-spaces. These theorems are proved under Martin’s axiom MA(ω 1). The results cannot be extended to spaces of weight ≥ω 2. For spaces of weight ω 1, these results cannot be obtained without additional set-theoretic assumptions. Thus, the question as to whether the space $$U_\tau (\mathbb{R}^{\omega _1 } )$$ is an absolute extensor cannot be answered in ZFC. The main result cannot be transferred to the functor U R of the unit ball of Radon measures. Indeed, the space $$U_R (\mathbb{R}^{\omega _1 } )$$ is not real-compact and, therefore, $$U_R (\mathbb{R}^{\omega _1 } ) \notin AE(0)$$ .