Real projective three-dimensional space

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

As a topological space, this can be defined in the following equivalent ways:

The real projective space of dimension 3. It is denoted $R P^{3}$ or $P^{3} R$ .
The underlying topological space of the special orthogonal group $S O (3, R)$ .

Note that real projective spaces are not in general homeomorphic to the underlying spaces of special orthogonal groups.

Topological space properties

Property	Satisfied?	Is the property a homotopy-invariant property of topological spaces?	Explanation	Corollary properties satisfied/dissatisfied
manifold	Yes	No	Based on the definition; in fact, any finite-dimensional real projective space is a manifold. We can also see this from the fact that its double cover, the 3-sphere, is a manifold	satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space	Yes	Yes	Can be seen directly, or from the fact that its double cover, the 3-sphere, is path-connected.	satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space	No	Yes	It has a double cover, namely the 3-sphere, which is path-connected. In fact, the double cover is simply connected, so the fundamental group of the space is a cyclic group of order two.	dissatisfies: weakly contractible space, contractible space
acyclic space	No	Yes	$H_{1}$ of the space is cyclic of order two (this can be seen from the Hurewicz theorem and the fact that $π_{1}$ is cyclic of order two, or directly using the homology of real projective space).	dissatisfies: weakly contractible space, contractible space
rationally acyclic space	No	Yes	The third homology group is isomorphic to $Z$	dissatisfies: space with Euler characteristic one
space with zero Euler characteristic	Yes	Yes	There are many ways of seeing this: Euler characteristic of compact connected odd-dimensional manifold is zero, Euler characteristic of compact connected nontrivial Lie group is zero (using the view of it as $S O (3, R)$ ), and the fact that its double cover is the 3-sphere (which has Euler characteristic zero) and Euler characteristic of covering space is degree of covering times Euler characteristic of base. It also follows from the homology of real projective space]]	-
compact space	Yes	No	Its double cover, the 2-sphere, is compact, and compactness is continuous image-closed	dissatisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness