Cone (topology)

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Need to create a book page, tidy up, and just generally flesh out the page, infobox may be nice too! We also may write $\frac{X\times I}{X\times\{1\} }$ for the equivalence relation, however nothing on this site mentions this alternate notation, probably because it isn't (eg $\frac{\mathbb{R} }{5\mathbb{Z } }$ being the $5$ equivalence classes where $[1]$ is all integers being concurrent to 1 mod 5 and such) Lastly, don't forget to check Mond's lecture notes on the subject

Definition

If $(X,\mathcal{J})$ is a topological space and $I$ denotes the unit interval, $[0,1]\subseteq\mathbb{R}$ (where $\mathbb{R}$ is considered with the usual topology on $\mathbb{R}$) the cone over $(X,\mathcal{J})$ is obtained by:

1. Constructing a new space, $X\times I$ (the product topology of $X$ and $I$)
2. Defining an equivalence relation, $\sim$ by:
   • For $(x,t),(x',t')\in X\times I$ we say $(x,t)\sim(x',t')$ if $t=t'=1$

     Notice we identify every point in $X\times\{1\}$ with every other point in $X\times\{1\}$ - this is the point of the cone.

3. The cone over $X$ is the quotient space $$\frac{X\times I}{\sim}$$