Given a topological space , the cone on (sometimes denoted by ) is the quotient space Note that there is a natural inclusion which sends to
If is a based topological space, there is a similar reduced cone construction, given by With this definition, the natural inclusion becomes a based map, where we take to be the basepoint of the reduced cone.
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![$ X\times [0,1] / (X\times \left\{0\right\})\cup(\left\{x_0\right\}\times [0,1]).$](http://images.planetmath.org:8080/cache/objects/3974/l2h/img9.png "$ X\times [0,1] / (X\times \left\{0\right\})\cup(\left\{x_0\right\}\times [0,1]).$"){kind=small}
