I don‘t understand the following argument in the paper „On the Steenrod homology theory“ by John Milnor: Let $(X,A)$ be a compact pair (I think compact includes Hausdorff here), $f: \Delta^p \times A \to S^n$ a continuous nullhomotopic map ($\Delta^p$ is the standard $p$-simplex). Then we can extend $f$ to a map $\Delta^p \times X \to S^n$. (The argument is on p. 92)
Why can we find such an extension? I tried using Tietze, but I couldn‘t find a solution. Also the cofibration property would be enough, but I doubt we have it here.
First note that $(X',A') = (\Delta^p \times X, \Delta^p \times A)$ is again a compact pair, thus it suffices to consider a continuous nullhomotopic map $f : A \to S^n$ and show that it extends to $X$.
More generally, let us consider a homotopy $h : A \times I \to S^n$ such that $h_0 = f$ and $h_1$ has an extension $H : X \to S^n$. We want to find an extension $F : X \to S^n$ of $f$. This looks like the usual homotopy extension property, but the problem is that $A \hookrightarrow X$ is in general not a cofibration.
However, $S^n$ is an ANR (= absolute neigborhood retract) and therefore Borsuk's homotopy extension theorem applies. See
You can read it here.