Tietze's extension theorem for contractible manifolds

Question

I've read that the Tietze's extension theorem was still valid for continuous applications from a closed subspace of a normal topological space to a contractible topological manifold (understood as Hausdorf and 2nd countable).

But I can't find any clear reference for this result.

What I have found is that the theorem generalize to applications from a normal space to an Absolute Retract (but for which family ? Normal spaces ? Metric spaces ? both ?), that manifolds are ANR (Abolute Neighbourhood Retract, once again for which family of spaces ?), and that Contractible ANR implies AR. Is this correct ?

Is there a direct proof somewhere that Tietze generalizes to contractible manifolds ?

PS: it is not a research-level question, but I can't get any answer on MathStackExchange... sorry.

Answer 1

Let us say that a topological space $Y$ is a normal absolute extensor if each continuous map $f:Z\to Y$ defined on a closed subspace $Z$ of a normal topological space $X$ extends to a continuous function $\bar f:X\to Y$. By the Tietze Theorem, the real line is a normal absolute extensor and so is the countable product $\mathbb R^\omega$ of real lines and any retract of $\mathbb R^\omega$.

Since each Polish (= separable completely metrizable) space admits a closed embedding into $\mathbb R^\omega$, we conclude that Polish AR's are normal absolute extensors. This fact is explicitely written as Theorem 16.1(d) in the old book ``Theory of Rectracts'' of S.-T. Hu.

Now it remains to observe that each contractible topological manifold $M$ is a Polish contractible ANR and hence a Polish AR. So, $M$ is a normal absolute extensor.