Real projective space: Difference between revisions

Revision as of 14:06, 2 April 2011

Definition

Short definition

Real projective space is defined as projective space over the topological field, namely, the field of real numbers.

Finite-dimensional

Real projective space of dimension $n$ , denoted $\mathbb {R} \mathbb {P} ^{n}$ or $\mathbb {P} ^{n}(\mathbb {R} )$ , is defined as the quotient space under the group action $\mathbb {R} ^{n+1}\setminus \{0\}/\mathbb {R} ^{*}$ where $\mathbb {R} ^{*}$ acts by scalar multiplication. It is equipped with the quotient topology.

As a set, we can think of it as the set of lines through the origin in $\mathbb {R} ^{n+1}$ . Using an inner product on $\mathbb {R} ^{n+1}$ , it can also be identified with the set of hyperplanes of codimension 1 (i.e., $n$ -dimensional linear subspaces) in $\mathbb {R} ^{n+1}$ .

Countable-dimensional

This space, called countable-dimensional real projective space and denoted $\mathbb {R} \mathbb {P} ^{\infty }$ , is defined as the quotient space of the nonzero elements of a countable-dimensional real vector space (with the standard topology) over $\mathbb {R}$ by the action of $\mathbb {R} ^{*}$ by scalar multiplication.

Others

We can also consider the real projective space corresponding to any topological real vector space, possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is $V$ , the projective space is defined as follows:we take $V\setminus \{0\}$ with the subspace topology, and then put the quotient topology on its quotient under the action of $\mathbb {R} ^{*}$ .

Particular cases

$n$	Real projective space $\mathbb {R} \mathbb {P} ^{n}$
0	one-point space
1	real projective line, which turns out to be homeomorphic to the circle
2	real projective plane
3	link: Fill this in later
countable ( $\infty$ )	countable-dimensional real projective space

Algebraic topology

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 ==Definition==
+===Short definition===
+'''Real projective space''' is defined as [[projective space]] over the [[topological field]], namely, the [[field of real numbers]].
 ===Finite-dimensional===
@@ Line 9: / Line 13: @@
 ===Countable-dimensional===
-This space, called [[countable-dimensional real projective space]] and denoted <math>\R\mathbb{P}^\infty</math>, is defined as the quotient space of the nonzero elements of a countable-dimensional vector space over <math>\R</math> by the action of <math>\R^*</math> by scalar multiplication.
+This space, called [[countable-dimensional real projective space]] and denoted <math>\R\mathbb{P}^\infty</math>, is defined as the quotient space of the nonzero elements of a [[countable-dimensional real vector space]] (with the standard topology) over <math>\R</math> by the action of <math>\R^*</math> by scalar multiplication.
+===Others===
+We can also consider the real projective space corresponding to any [[topological real vector space]], possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is <math>V</math>, the projective space is defined as follows:we take <math>V \setminus \{ 0 \}</math> with the [[subspace topology]], and then put the [[quotient topology]] on its quotient under the action of <math>\R^*</math>.
 ==Particular cases==