# Earliest use of adjunction spaces?

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.

• 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) – Ulli Jan 13 at 21:09
• Thanks a lot:.I've just printed it out! – Ronnie Brown Jan 13 at 21:49
• @Ulli Why not an official answer? – Paul Frost Jan 13 at 22:57
• The information is in the second edition of Engilking's book! – Ronnie Brown Jan 17 at 15:03