## You are here

Homecone

## Primary tabs

# cone

Given a topological space $X$, the cone on $X$ (sometimes denoted by $CX$) is the quotient space $X\times[0,1]/X\times\left\{0\right\}.$ Note that there is a natural inclusion $X\hookrightarrow CX$ which sends $x$ to $(x,1).$

If $(X,x_{0})$ is a based topological space, there is a similar reduced cone construction, given by $X\times[0,1]/(X\times\left\{0\right\})\cup(\left\{x_{0}\right\}\times[0,1]).$ With this definition, the natural inclusion $x\mapsto(x,1)$ becomes a based map, where we take $(x_{0},0)$ to be the basepoint of the reduced cone.

Defines:

reduced cone

Related:

Suspension, Join3

Major Section:

Reference

Type of Math Object:

Definition

## Mathematics Subject Classification

54B99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff