nLab space attachment




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



In topology, the result of a space attachment (sometimes called an attaching space or adjunction space) is a topological space, denoted X fYX \cup_{f} Y, which is constructed by “attaching” or “gluing” two topological spaces XX and YY along a topological subspace AXA \subset X by means of a continuous function f:AYf \colon A \to Y. The function ff is then called the attaching map.

(graphics taken from AGP08, §3.1)

More abstractly, space attachments are pushouts along monomorphisms in the category Top of all topological spaces. The formally dual concept is that of fiber spaces or more generally of fiber products of topological spaces.


Let X,YTopX,Y \in Top be topological spaces, let AXA \subset X be a topological subspace and let f:AYf \colon A \to Y be a continuous function.

Then the attaching space X fYTopX \cup_f Y \in Top may be realized as the quotient topological space of the disjoint union space XYX \sqcup Y by the equivalence relation which identifies a point xAXx \in A \subset X with its image f(x)Yf(x) \in Y:

X fY(XY)/. X \cup_f Y \;\simeq\; \left( X \sqcup Y \right)/\sim \,.

More category theoretically, the attaching space is the pushout in the category Top of topological spaces of the subspace inclusion i:AXi \colon A \hookrightarrow X along ff, i.e. the topological space which is universal with the property that it makes the following square commute:

A AiA X f (po) Y X fY. \array{ A &\overset{\phantom{A}i\phantom{A}}{\hookrightarrow}& X \\ {}^{\mathllap{f}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \cup_f Y } \,.

For more on this see at Top – Universal constructions.


examples of universal constructions of topological spaces:

\, point space\,\, empty space \,
\, product topological space \,\, disjoint union topological space \,
\, topological subspace \,\, quotient topological space \,
\, fiber space \,\, space attachment \,
\, mapping cocylinder, mapping cocone \,\, mapping cylinder, mapping cone, mapping telescope \,
\, cell complex, CW-complex \,


See also

Last revised on February 11, 2023 at 13:14:20. See the history of this page for a list of all contributions to it.