On Borsuk’s Non-Retract Theorem

Let X be a Hausdorff topological space and let A be both an F σ and G δ subset of X .Let also f : A → ℝ be a function for which the inverse image of every open subset U ⊂ ℝ is F σ in A . We will prove that there is a linear extension operator ϕ * such that ϕ * ( f ) has the same property on X . An analogous result is proved for the Baire-one function defined on an analogous subset of ℝ. We will also show that the extension map is (with a supremum norm) an isometry. In the second part of the paper, we deal with classical Borsuk’s non-retract theorem. It says that a unit sphere in ℝ n is not a continuous retract of the unit closed ball. We will show that such a unit sphere is a piecewise continuous retract of the unit closed ball.