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 ℝⁿ 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.