# 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 Cone 2013-03-22 13:25:20 2013-03-22 13:25:20 antonio (1116) antonio (1116) 7 antonio (1116) Definition msc 54B99 Suspension Join3 reduced cone