Difference between revisions of "Exercises:Mond - Topology - 1/Question 7"
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==Section B== | ==Section B== | ||
===Question 7=== | ===Question 7=== | ||
| − | </noinclude>Let {{M|D^2}} denote the [[closed unit disk]] in {{M|\mathbb{R}^2}} and define an [[equivalence relation]] on {{M|D^2}} by setting {{M|1=x_1\sim x_2}} if {{M|1=\Vert x_1\Vert=\Vert x_2\Vert}}. Show that {{M|\frac{D^2}{\sim} }} is [[homeomorphic]] to {{M|\mathbb{S}^2}} - the [[sphere]]. | + | </noinclude>Let {{M|D^2}} denote the [[closed unit disk]] in {{M|\mathbb{R}^2}} and define an [[equivalence relation]] on {{M|D^2}} by setting {{M|1=x_1\sim x_2}} if {{M|1=\Vert x_1\Vert=\Vert x_2\Vert=1}} ("collapsing the boundary to a single point"). Show that {{M|\frac{D^2}{\sim} }} is [[homeomorphic]] to {{M|\mathbb{S}^2}} - the [[sphere]]. |
* '''Hint: ''' first define a [[surjection]] {{M|(:D^2\rightarrow\mathbb{S}^2)}} mapping all of {{M|\partial D^2}} to the north pole. This may be defined using a good picture or a formula. | * '''Hint: ''' first define a [[surjection]] {{M|(:D^2\rightarrow\mathbb{S}^2)}} mapping all of {{M|\partial D^2}} to the north pole. This may be defined using a good picture or a formula. | ||
====Solution==== | ====Solution==== | ||
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==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Revision as of 11:56, 8 October 2016
Section B
Question 7
Let [ilmath]D^2[/ilmath] denote the closed unit disk in [ilmath]\mathbb{R}^2[/ilmath] and define an equivalence relation on [ilmath]D^2[/ilmath] by setting [ilmath]x_1\sim x_2[/ilmath] if [ilmath]\Vert x_1\Vert=\Vert x_2\Vert=1[/ilmath] ("collapsing the boundary to a single point"). Show that [ilmath]\frac{D^2}{\sim} [/ilmath] is homeomorphic to [ilmath]\mathbb{S}^2[/ilmath] - the sphere.
- Hint: first define a surjection [ilmath](:D^2\rightarrow\mathbb{S}^2)[/ilmath] mapping all of [ilmath]\partial D^2[/ilmath] to the north pole. This may be defined using a good picture or a formula.
Solution
Comments:
- Suppose we take the hind and find a surjection, [ilmath]f:D^2\rightarrow\mathbb{S}^2[/ilmath], what would we do next? Passing to the quotient again! Then, as already mentioned, invoke the compact-to-Hausdorff theorem to yield a homeomorphism
Notes
References
