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## Main Question or Discussion Point

In the first volume of Differential Geometry, Ch. 1, Spivak states that if [itex]U \subset \mathbb{R}^n[/itex] is homeomorphic to [itex]\mathbb{R}^n[/itex], then [itex]U[/itex] is open. This seems obvious: [itex]\mathbb{R}^n[/itex] is open in [itex]\mathbb{R}^n[/itex], so its pre-image under a homeomorphism [itex]f:U \rightarrow \mathbb{R}^n[/itex] is open. The pre-image under [itex]f[/itex] of [itex]\mathbb{R}^n[/itex] is [itex]U[/itex]. Therefore [itex]U[/itex] is open in [itex]\mathbb{R}^n[/itex].

Why does Spivak not take this obvious route? Am I mistaken about it? Instead, he says that proof of the openness of [itex]U[/itex] needs something called the Invariance of domain theorem.

Why does Spivak not take this obvious route? Am I mistaken about it? Instead, he says that proof of the openness of [itex]U[/itex] needs something called the Invariance of domain theorem.