This article is about a particular topological space (uniquely determined up to homeomorphism)View a complete list of particular topological spaces
Definition
This topological space is defined as the Cartesian product of the 2sphere and the real projective plane equipped with the product topology. It is denoted or as .
Topological space properties
Property 
Satisfied? 
Is the property a homotopyinvariant property of topological spaces? 
Explanation 
Corollary properties satisfied/dissatisfied

manifold 
Yes 
No 
product of manifolds is manifold 
satisfies: metrizable space, secondcountable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.

pathconnected space 
Yes 
Yes 
pathconnectedness is productclosed 
satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)

simply connected space 
No 
Yes 
Product of two spaces, one of which (the real projective plane) is not simply connected. 
dissatisfies: simply connected manifold

rationally acyclic space 
No 
Yes 
The 2sphere is not rationally acyclic. 
dissatisfies: acyclic space, weakly contractible space, contractible space

space with Euler characteristic zero 
No 
Yes 
Product of two spaces, neither of which has Euler characteristic zero. Note that Euler characteristic of product is product of Euler characteristics 

space with Euler characteristic one 
No 
Yes 
The Euler characteristic is . 

compact space 
Yes 
No 
Product of compact spaces, see Tychonoff's theorem 
satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

Algebraic topology
Homology groups
Further information: Kunneth formula, homology of spheres, homology of real projective space
The homology groups with coefficients in integers are as follows: