Difference between revisions of "Cone (topology)"
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Latest revision as of 21:44, 20 April 2016
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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 [ilmath]\frac{X\times I}{X\times\{1\} } [/ilmath] for the equivalence relation, however nothing on this site mentions this alternate notation, probably because it isn't (eg [ilmath]\frac{\mathbb{R} }{5\mathbb{Z } } [/ilmath] being the [ilmath]5[/ilmath] equivalence classes where [ilmath][1][/ilmath] 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 [ilmath](X,\mathcal{J})[/ilmath] is a topological space and [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subseteq\mathbb{R} [/ilmath] (where [ilmath]\mathbb{R} [/ilmath] is considered with the usual topology on [ilmath]\mathbb{R} [/ilmath]) the cone over [ilmath](X,\mathcal{J})[/ilmath] is obtained by[1]:
- Constructing a new space, [ilmath]X\times I[/ilmath] (the product topology of [ilmath]X[/ilmath] and [ilmath]I[/ilmath])
- Defining an equivalence relation, [ilmath]\sim[/ilmath] by:
- For [ilmath](x,t),(x',t')\in X\times I[/ilmath] we say [ilmath](x,t)\sim(x',t')[/ilmath] if [ilmath]t=t'=1[/ilmath]
- Notice we identify every point in [ilmath]X\times\{1\} [/ilmath] with every other point in [ilmath]X\times\{1\} [/ilmath] - this is the point of the cone.
- For [ilmath](x,t),(x',t')\in X\times I[/ilmath] we say [ilmath](x,t)\sim(x',t')[/ilmath] if [ilmath]t=t'=1[/ilmath]
- The cone over [ilmath]X[/ilmath] is the quotient space [math]\frac{X\times I}{\sim} [/math]
References
- ↑ An Introduction to Algebraic Topology - Joseph J. Rotman
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