# Homeomorphism between product of spheres and euclidean space

I need to prove that ${S^n} \times {S^k}$ is homeomorphic to a subspace of ${\mathbb{R}^{n + k + 1}}$ by constructing an explicit map between the two.

I am unsure how to start this as I can't seem to think of any suitable maps. A spherical coordinate type map does not have a continuous inverse and a stereographic projection type map leaves out a point on each sphere.

• Use the typical realization of the torus ($S^1\times S^1$) in $\mathbb{R}^3$ as an inspiration. – user147263 Feb 8 '16 at 0:50
• I am not familiar with the proof to go along with that. Any good resource to see it? – topology123 Feb 8 '16 at 0:54
• @topology123 : I'd try first actually constructing the thing suggested by "Live Forever" and then afterwards think about proving that you have a homeomorphism. I'd guess that in the process of constructing the torus-like object you'd construct the homeomorphism; after that you'd only need to prove that it's a homeomorphism. The way to write the proof should be suggested by the way the construction is done. $\qquad$ – Michael Hardy Feb 8 '16 at 1:07
• @topology123: just google for "torus"for online resources. Then think how the surface obtained by rotating a circle about a circle (in $\Bbb{R}^3$) could generalise to rotating an $n$-sphere around a $k$-sphere (in $\Bbb{R}^{n+k+1}$). – Rob Arthan Feb 8 '16 at 1:23
• I am not really finding anything showing how the torus defined by S1xS1 is a subspace of R^3. – topology123 Feb 8 '16 at 1:53