TITLE: Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?
QUESTION [6 upvotes]: It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$
is necessary to make $\left(X,A\right)$ a pair having the homotopy
extension property (HEP). In Note on cofibrations written in 1966
by Arne Strøm I meet the remark that it is sufficient
under the extra condition that $A$ is closed. In Hatcher's Algebraic
Topology it is shown that this extra condition can be dropped. Who
was the first to find that out, and when did that happen? 

My last comment is not immediately visible on my screen and I do not
want to miss this opportunity. So here I repeat it (probably unnecessary)
as an addition of my original question. Indeed, as professor Brown remarked, that is what the
debate is about: if $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$
is a retract of $X\times\mathbb{I}$ then the pair $\left(X\times\left\{ 0\right\} ,A\times\mathbb{I}\right)$
has the gluing property. Professor Brown, I would like to attend you
on a nice consequence of the proof. In Topology and Groupoids (nice
book!) 7.2.4 it is shown that $1_{B}\times i$ is a cofibration if
$i$ is, but this under the extra condition that $B$ is locally compact.
This extra condition can be left out! If $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$
is a retract of $X\times\mathbb{I}$ then $B\times X\times\left\{ 0\right\} \cup B\times A\times\mathbb{I}$
is a retract of $B\times X\times\mathbb{I}$, and that is enough.

REPLY [13 votes]: This result is proved in a second paper by Arne Strøm, "Note on Cofibrations II" in Math. Scand. 22 (1968), 130-142.  As far as I am aware this is the original source, and in any case it was the source for the proof given in the appendix of my book (added in 2009 to the online version of the book). I will add this paper to the list of references in the book -- I don't know why I didn't do this before.