# Difference between revisions of "Cone"

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− | Let $(X,\tau)$ be a [[topological space]]. We define the cone over $X$ taking $ | + | Let $(X,\tau)$ be a [[topological space]]. We define the cone over $X$ taking $X \times [0,1]$ and shrinking $X \times \{1\}$ to a point. More precisely we define $\mathrm{Cone}(X)$ to be the [[quotient space]] $X \times [0,1] / \sim$, where $\sim$ is an [[equivalence class]] defined by $(x_1,t_1) \sim (x_2,t_2)$ if and only if $t_1=t_2=1$. The point $X \times \{1\} \in \mathrm{Cone}(X)$ is called the vertex of the cone while $X \times \{0\}$ is called the base of $\mathrm{Cone}(X)$. |

## Revision as of 07:54, 5 April 2015

Let $(X,\tau)$ be a topological space. We define the cone over $X$ taking $X \times [0,1]$ and shrinking $X \times \{1\}$ to a point. More precisely we define $\mathrm{Cone}(X)$ to be the quotient space $X \times [0,1] / \sim$, where $\sim$ is an equivalence class defined by $(x_1,t_1) \sim (x_2,t_2)$ if and only if $t_1=t_2=1$. The point $X \times \{1\} \in \mathrm{Cone}(X)$ is called the vertex of the cone while $X \times \{0\}$ is called the base of $\mathrm{Cone}(X)$.