It is proved that for every Hausdorff space ℝ and for every Hausdorff (regular or Moore) space X, there exists a Hausdorff (regular or Moore, respectively) space S containing X as a closed subspace and having the following properties:
la) Every continuous map of S into ℝ is constant.
b) For every point x of S and every open neighbourhood U of x there exists an open neighbourhood V of x, V ⊆ U such that every continuous map of V into ℝ is constant.
2) Every continuous map f of S into S (f ≠ identity on S) is constant.
In addition it is proved that the Fomin extension of the Moore space
S has these properties.
