nLab
space attachment

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

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.

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.

Definition

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:

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

For more on this see at Top – Universal constructions.

Examples

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\phantom{AAAA}colimits
\, 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 \,

References

Revised on May 21, 2017 08:49:45 by Urs Schreiber (92.218.150.85)