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# 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*

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