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          <rev contentformat="text/x-wiki" contentmodel="wikitext" xml:space="preserve">==Definition==
A ''[[function]]'' is said to be ''real-valued'' if the [[co-domain]] is the set of [[real numbers]], {{M|\mathbb{R} }}&lt;ref name=&quot;ITSM&quot;&gt;Introduction to Smooth Manifolds - Second Edition - John M. Lee - Springer GTM&lt;/ref&gt;. That is to say any function ( {{M|f}} ) and any set ( {{M|U}} ) such that:
* {{M|f:U\rightarrow\mathbb{R} }}
==See also==
* [[Extended-real-valued function]]
* [[Extended real value|Extended-real-value]]
* [[Class of smooth real-valued functions on R-n|The class of smooth real-valued functions on {{M|\mathbb{R}^n}}]]
* [[Class of k-differentiable real-valued functions on R-n|The class of {{M|k}}-differentiable real-valued functions on {{M|\mathbb{R}^n}}]]
==References==
&lt;references/&gt;
{{Definition|Measure Theory|Manifolds|Differential Geometry|Functional Analysis}}</rev>
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      <page pageid="2026" ns="0" title="Real projective space">
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          <rev contentformat="text/x-wiki" contentmodel="wikitext" xml:space="preserve">{{Stub page|grade=A*|msg=Would be a great page to have
* Demote to grade C once charts and definition 1 is in place [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 06:21, 18 February 2017 (UTC)}}
__TOC__
==Definition==
Let {{M|n\in\mathbb{N}_{\ge 1} }} be given. There are 2 common definitions for {{M|\mathbb{RP}^n}} that we encounter. We will use definition 1 unless otherwise noted throughout the unified mathematics project.
===Definition 1===
{{Infobox|style=max-width:25em;
|title=Definition 1
|above=&lt;span style=&quot;font-size:1.6em;&quot;&gt;&lt;mm&gt;\frac{\mathbb{S}^n\subset\mathbb{R}^{n+1} }{\langle x\sim -x\rangle}&lt;/mm&gt;&lt;/span&gt;
}}
&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;
===Definition 2===
{{Infobox|style=max-width:25em;
|title=Definition 2
|above=&lt;span style=&quot;font-size:1.1em;&quot;&gt;{{MM|\frac{\mathbb{R}^{n+1}-\{0\} }{\langle x\sim\lambda x\ \vert\ \lambda\in(\mathbb{R}-\{0\})\rangle} }}&lt;/span&gt;
}}
* {{M|\mathbb{RP}^n:\eq\{L\in\mathcal{P}(\mathbb{R}^{n+1})\ \vert\ (L,\mathbb{R})\text{ is an 1-} }}[[dimension (vector space)|{{M|\text{dimensional} }}]]{{M|\text{ vector } }}[[vector subspace|{{M|\text{subspace} }}]]{{M|\text{ of }(\mathbb{R}^{n+1},\mathbb{R})\} }}
Of course doesn't tell us what [[topology]] to consider {{M|\mathbb{RP}^n}} with, for that, define the [[map]]:
* {{M|\pi:(\mathbb{R}^{n+1}-\{0\})\rightarrow\mathbb{RP}^n}} given by: {{M|\pi:x\mapsto\langle x\rangle}}
** We use this map to imbue {{M|\mathbb{RP}^n}} with the [[quotient topology]], so:
*** {{MM|\mathbb{RP}^n\cong\frac{\mathbb{R}^{n+1}-\{0\} }{\pi} }} {{XXX|What does this actually mean though? In terms of quotient-ing by an [[equivalence relation]]!}}
&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;
==Named instances==
* [[Real projective plane]] - {{M|\mathbb{RP}^2}}

==Standard structure==
===As a [[Topological manifold|topological {{n|manifold}}]]===
{{Requires work|grade=A*|msg=Charts}}
==References==
&lt;references/&gt;
{{Definition|Smooth Manifolds|Topological Manifolds|Manifolds}}</rev>
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