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, equipped with the product topology, of the 2sphere and the 2torus (which itself is the product of two copies of the circle ). It is denoted or or or .
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 multiple spaces, one of which (the circle) is not simply connected. 
dissatisfies: simply connected manifold

rationally acyclic space 
No 
Yes 
Product of spaces neither of which is rationally acyclic. 
dissatisfies: acyclic space, weakly contractible space, contractible space

space with Euler characteristic zero 
Yes 
Yes 
Product of two spaces, one of which (the 2torus) has Euler characteristic zero. Note that Euler characteristic of product is product of Euler characteristics 

space with Euler characteristic one 
No 
Yes 
See above, the Euler characteristic is zero 

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
