Cone

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Let $(X,\tau)$ be a topological space. We define the cone over $X$ taking $Y \times [0,1]$ and shrinking $Y \times \{1\}$ to a point. More precisely we define $\mathrm{Cone}(X)$ to be the quotient space $Y \times [0,1] / \sim$, where $\sim$ is an equivalence class defined by $(y_1,t_1) \sim (y_2,t_2)$ if and only if $t_1=t_2=1$.