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I started to reread Spivak's

Embarrassingly, I did not find it quite as easy as I'd have hoped. Flustered, I tried looking on the internet. I found various proofs of the statement that "the neighborhood U in our definition [of a manifold] must be open." I found one that was particularly easy to follow (All the proofs I saw certainly weren't what I'd call "easy exercises."), but I didn't understand the last step. The proof-writer explains his or her last step as "We have proved that any point x in U has an open neighborhood W contained in U, therefore U is open."

Can someone explain why this is true? I'm probably just missing something really simple here.

Additionally, would you consider the proof that "the neighborhood U in our definition [of a manifold] must be open" to be trivial? I certainly don't see the "easy exercise" Spivak must have envisioned.

Thank you.

*A Comprehensive Introduction to Differential Geometry*last night, for the sake of attempting to improve my ability to do differential geometry. I noticed that I had skipped what Spivak calls an "easy exercise" after he introduces Invariance of Domain.Embarrassingly, I did not find it quite as easy as I'd have hoped. Flustered, I tried looking on the internet. I found various proofs of the statement that "the neighborhood U in our definition [of a manifold] must be open." I found one that was particularly easy to follow (All the proofs I saw certainly weren't what I'd call "easy exercises."), but I didn't understand the last step. The proof-writer explains his or her last step as "We have proved that any point x in U has an open neighborhood W contained in U, therefore U is open."

Can someone explain why this is true? I'm probably just missing something really simple here.

Additionally, would you consider the proof that "the neighborhood U in our definition [of a manifold] must be open" to be trivial? I certainly don't see the "easy exercise" Spivak must have envisioned.

Thank you.

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