To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Adjunction space

From Wikipedia, the free encyclopedia

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : AX be a continuous map (called the attaching map). One forms the adjunction space Xf Y (sometimes also written as X +f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally,

where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, Xf Y consists of the disjoint union of X and (YA). The topology, however, is specified by the quotient construction.

Intuitively, one may think of Y as being glued onto X via the map f.


  • A common example of an adjunction space is given when Y is a closed n-ball (or cell) and A is the boundary of the ball, the (n−1)-sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex.
  • Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from X and Y before attaching the boundaries of the removed balls along an attaching map.
  • If A is a space with one point then the adjunction is the wedge sum of X and Y.
  • If X is a space with one point then the adjunction is the quotient Y/A.


The continuous maps h : Xf YZ are in 1-1 correspondence with the pairs of continuous maps hX : XZ and hY : YZ that satisfy hX(f(a))=hY(a) for all a in A.

In the case where A is a closed subspace of Y one can show that the map XXf Y is a closed embedding and (YA) → Xf Y is an open embedding.

Categorical description

The attaching construction is an example of a pushout in the category of topological spaces. That is to say, the adjunction space is universal with respect to the following commutative diagram:

Here i is the inclusion map and ϕX, ϕY are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of X and Y. One can form a more general pushout by replacing i with an arbitrary continuous map g—the construction is similar. Conversely, if f is also an inclusion the attaching construction is to simply glue X and Y together along their common subspace.

See also


  • Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts. (Provides a very brief introduction.)
  • "Adjunction space". PlanetMath.
  • Ronald Brown, "Topology and Groupoids" pdf available  , (2006) available from amazon sites. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes.
  • J.H.C. Whitehead "Note on a theorem due to Borsuk" Bull AMS 54 (1948), 1125-1132 is the earliest outside reference I know of using the term "adjuction space".
This page was last edited on 23 October 2021, at 13:48
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.