How to prove the Cone is contractible?

Question

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are:

• $A$ itself and
• Singletons $\{x\}$ such that $x ∉ A$.

Then define $X/A$ to be the quotient space $X/{\sim}$. (i.e. collapse $A$ to a point)

The cone on $X$, denoted $CX$, is the quotient space $(X × [0, 1]) / (X × ${$0$}$)$.

How can I prove that $CX$ is contractible? (by this I mean that it is homotopic to a constant map, intuitively that it can be continuously shrunk to a point)

Answer 1

First consider the map $H : X \times [0,1] \times [0,1] \to X \times [0,1]$ defined by $H((x,t),s) = (x,(1-s)t)$. Then $H$ is continuous and $H((x,0),s) = (x,0)$ for all $x\in X$ and $s\in [0,1]$. Using these facts, argue that $H$ induces a continuous map $\hat{H} : CX \times [0,1] \to CX$ such that $\hat{H}([(x,t)],s) = [(x,(1-s)t)]$. Now $\hat{H}([(x,t)],0) = [(x,t)]$ and $\hat{H}([(x,t)],1) = [(x,0)] = X \times \{0\}$. So $\hat{H}$ is a homotopy in $CX$ from the identity on $CX$ to the point $X \times \{0\}$. Consequently, $CX$ is contractible.