View source for Exercises:Mond - Topology - 1/Question 9

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==Section B==
===Question 9===</noinclude>
The [[real projective plane]], {{M|\mathbb{RP}^2}} is defined as the [[quotient topology|quotient]] of the [[sphere]], {{M|\mathbb{S}^2}}, by the [[equivalence relation]] that defines (for {{M|x\in\mathbb{S}^2\subset\mathbb{R}^3}}) {{M|x\sim -x}}, that is it identifies antipodal points.

Show that {{M|\mathbb{RP}^2}} is [[Hausdorff]]
====Definitions====
* We denote by {{M|\pi:\mathbb{S}^2\rightarrow\frac{\mathbb{S}^2}{\sim} }} the [[canonical projection of the equivalence relation]], {{M|\sim}}. Note that this is a {{link|quotient map|topology}} when we consider {{M|\frac{\mathbb{S}^2}{\sim} }} with the [[quotient topology]].
====Solution outline====
[[File:RP2isHausdorff.JPG|thumb|Diagram...]]We will deal with the [[open sets]], {{M|U}}, in terms of {{M|\pi^{-1}(U)}} (as by definition, {{M|U\in\mathcal{P}(\frac{\mathbb{S}^2}{\sim})}} is open in {{M|1=\frac{\mathbb{S}^2}{\sim} }} {{iff}} {{M|\pi^{-1}(U)}} is open in {{M|\mathbb{S}^2}}, which we consider with the [[subspace topology]] inherited from {{M|\mathbb{R}^3}} as usual) then we just have to find small enough open balls!

With this in mind, let {{M|a,b\in\mathbb{RP}^2}} be given. We need to find two [[open sets]] (well... [[neighbourhoods]] will do... but open sets are neighbourhoods!), let's say {{M|U_a}} and {{M|U_b}} such that:
* {{M|1=a\in U_a}}, {{M|b\in U_b}} and {{M|1=U_a\cap U_b=\emptyset}}
Well:
* {{M|U_a}} is [[open set|open]] in {{M|\mathbb{RP}^2}} 
{{iff}}
* {{M|\pi^{-1}(U_a)}} is open in {{M|\mathbb{S}^2}} 
{{iff}} 
* there exists an [[open set]], {{M|V_a}} in {{M|\mathbb{R}^3}} such that {{M|1=V_a\cap\mathbb{S}^2=\pi^{-1}(U_a)}}
Of course, the [[open balls]] of {{M|\mathbb{R}^3}} are a {{link|basis|topology}}, so we can think of {{M|V_a}} as a union of open balls, or possibly just an open ball (as basis sets themselves are open, and also as an open ball can be expressed as a union of open balls). 

This changes the question into, in terms of {{M|a}} and {{M|b}}, what size balls can we consider in {{M|\mathbb{R}^3}} such that they're disjoint. There's a caveat here. This is what is shown in the diagram.

If {{M|a}} and {{M|b}} are "far apart" on {{M|\mathbb{RP}^2}}, it is entirely possible (in the {{link|pre-image|map}} under {{M|\pi}}) that the antipodal point of one is near the other!

So we must be careful to make sure our balls do not overlap at all!
====Solution====
We wish to show that {{M|\mathbb{RP}^2}} is [[Hausdorff]].
* Let {{M|a,b\in\mathbb{RP}^2}} be given such that {{M|1=a\ne b}}, then
** there exist {{M|1=x,y\in\mathbb{S}^2}} such that {{M|1=\pi^{-1}(a)=\{x,-x\} }} and {{M|1=\pi^{-1}(b)=\{y,-y\} }} 
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==Notes==
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==References==
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