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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)

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  • $\begingroup$ Let $f:X\to Y$ be a continuous function, then how about the function $H:X\times [0,1]\to Y$, $H((x_0,h),r)=f(x_0,hr)$ where $x_0\in X$, $h,r\in[0,1]$? $\endgroup$
    – Hanul Jeon
    Dec 13, 2014 at 1:47
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    $\begingroup$ Try to come up with an intuitive argument (e.g. a drawing) for a simple case, e.g. if $X = \{1,2,\dots,10\}$ with the discrete topology. What is $CX$ in that case? Can you draw it? How would you contract it to a point? As to your second question: If seems that somebody might have considered this before. So perhaps there is a general result. Check your book or your notes. $\endgroup$ Dec 13, 2014 at 1:47

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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.

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