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An adjunction space is a topological space $Y= B \cup _f X$ determined by an inclusion $i : A \to X$ and a map $f: A \to B$; it can be described briefly as the pushout of the two maps $f,i$. The advantage of this notion is that it is behind the J.H.C. Whitehead notion of CW-complex, which is about inductively attaching cells $E^n$ by means of maps $S^{n-1} \to X_{n-1}$. Good properties of $A,X,B$ are often carried over to $Y$ (eg being Hausdorff). The pushout property allows one to construct continuous functions on $Y$. { cf Topology and Groupoids).

The earliest paper I have on this notion is by Whitehead: "Note on a Theorem due to Borsuk", Bull AMS (1948) 1125-1137. I would be grateful for information on earlier reference to this notion.

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    $\begingroup$ According to Engelking, General Topology, Historical and bibliographic notes of section 2.4, adjunction spaces were defined by Borsuk in 1935 (for compact metric spaces) $\endgroup$ – Ulli Jan 13 at 21:09
  • $\begingroup$ Thanks a lot:.I've just printed it out! $\endgroup$ – Ronnie Brown Jan 13 at 21:49
  • $\begingroup$ @Ulli Why not an official answer? $\endgroup$ – Paul Frost Jan 13 at 22:57
  • $\begingroup$ The information is in the second edition of Engilking's book! $\endgroup$ – Ronnie Brown Jan 17 at 15:03
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Since I was asked to state an official answer, here it is, although it's just copy paste from Engelking's book:

According to Engelking, General Topology, Historical and bibliographic notes of section 2.4, adjunction spaces were defined by Borsuk in 1935 (for compact metric spaces)

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