# Reference for the proof of a neighbourhood characterisation of cofibrations

I am interested in a reference for the proof of the following theorem for $A,X$ being CGWH topological spaces.

Let $A\subset X$ be a closed subspace, such that there exists a continuous $\phi : X \rightarrow I$ with $A=\phi^{-1}(0)$. Let $U=\phi^{-1}([0,1))$.

Suppose there exists a homotopy $h:U \times I \rightarrow X$, relative to $A$ (i.e. $g(a,t)=a$ for all $a\in A$ and $t \in I$), satisfying moreover the conditions:

• $g(x,0)=x$ for all $x\in U$
• $g(x,1) \in A$ for all $x\in U$

Then the inclusion $A\subset X$ is a cofibration.

I am also interested in any proof working a priori just for locally compact spaces, in which the fact of being locally compact appears only as a justification of properties of map spaces, as the proof may then work for CGWH spaces, too.

I would be especially glad to see a proof without any sort of "hand-waving" or omissions of form "it's easy to see the map is continuous". I have already managed to be discouraged to look for justifications of such claims by myself:

In Postnikov's "Lectures in algebraic topology: Elements of homotopy theory" (1984, in Russian) I found a proof of this statement, but it happened to have a flaw (some map claimed to be continuous, without any argument to it, wasn't necessarily such) and after some little modification it still wasn't clear to me it would work without additional assumption of $X$ being locally compact (but with this assumtpion the modification seems to make the reasoning valid).

I have looked up another proof, now not sure of where to find it again, which suffered from the same kind of omission (an unproven claim that some map is continuous, which happened to be false).

• Try also 7.3.7 of "Topology and Groupoids". A continuity proof is given as 7.3.10: Let $\phi: X \times C \to \mathbb R$ be continuous, let $C$ be compact, and let $w: X \to \mathbb R$ be $x \mapsto \sup _{c \in C} \phi(x,c)$. Then $w$ is well defined and continuous. – Ronnie Brown Jan 12 '15 at 10:45

Maybe the other proof you found is Theorem 2 of

Strøm, Arne Note on cofibrations. Math. Scand. 19 1966 11–14.

This deals with completely general spaces. Unfortunately the last line of the proof given is "The proof of continuity is straightforward and is omitted".

The MR review for that article mentions that the same result is proved in

Puppe, Dieter Bemerkungen über die Erweiterung von Homotopien. Arch. Math. (Basel) 18 1967 81–88.

which may contain more details (but is in German), and also some mimeographed notes of Puppe.

The result also appears as Exercise E6 in Chapter 1 of Spanier's text, where it is attributed to

Young, G.S. A condition for the absolute homotopy extension property. Amer. Math. Monthly 71 1964 896-897.

I realize this doesn't fully answer your question, but might give you a few more places to look.

You can find a detailed proof in May's book "A concise course in algebraic topology", in the chapter on cofibrations.