Singular Homology, Each Euclidean Space is Distinct

Each Euclidean Space is Distinct

Is the plane really different from 3 space? Or is there a homeomorphism that equates the two? Both sets have the same cardinality, so there is a function, a bijection, that maps R² onto R³. Is there a bijection that preserves open sets? There is not, because the spaces are topologically distinct.

Suppose f is a homeomorphism between R² and R³. Take the compactification of the domain and the range to get S² and S³ respectively. Let f map ω to ω. Verify that f is still a homeomorphism. Now the two spheres are homeomorphic. This has been ruled out by the previous theorem. Therefore each euclidean space Rⁿ is distinct.

Since Rⁿ maps homeomorphically onto the interior of Dⁿ, open balls of different dimensions are also topologically distinct.