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.

Title	cone
Canonical name	Cone
Date of creation	2013-03-22 13:25:20
Last modified on	2013-03-22 13:25:20
Owner	antonio (1116)
Last modified by	antonio (1116)
Numerical id	7
Author	antonio (1116)
Entry type	Definition
Classification	msc 54B99
Related topic	Suspension
Related topic	Join3
Defines	reduced cone