# Finishing a Supposedly Simple Proof

• Mandelbroth
In summary, the conversation discusses the difficulty of an "easy exercise" in Spivak's A Comprehensive Introduction to Differential Geometry, the search for an explanation on the internet, and a discussion on the proof that "the neighborhood U in our definition of a manifold must be open." The conversation ends with the person understanding the proof after initially misreading it.
Mandelbroth
I started to reread Spivak's 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.

Last edited:
If the set is closed there will be points (on the boundary) which are not in open sets ... because they don't have any neighbors beyond the boundary.

Think in terms of topology here.

UltrafastPED said:
If the set is closed there will be points (on the boundary) which are not in open sets ... because they don't have any neighbors beyond the boundary.
I...don't follow how this shows ##U## is open. :uhh:

Perhaps I did not understand your specific question. Please quote the actual exercise, including the definitions which have been provided.

UltrafastPED said:
Perhaps I did not understand your specific question. Please quote the actual exercise, including the definitions which have been provided.
The quoted part of the text is attached.

I consider a manifold to be a second countable Hausdorff space such that every point in the space has a neighborhood homeomorphic to ##\mathbb{R}^n##.

I'm trying to understand a proof by someone else, who finishes their proof with the phrase "We have proved that any point x in U has an open neighborhood W contained in U, therefore U is open."

Edit: Never mind. I figured it out. I read it as "a point x" rather than "any point x."

#### Attachments

• spivak1.JPG
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Glad you were able to figure it out!

## 1. What is the best approach to finishing a supposedly simple proof?

The best approach to finishing a supposedly simple proof is to start by identifying the key concepts and definitions involved. Then, break down the proof into smaller, more manageable steps and work through them systematically. It is also helpful to consult with peers or a mentor for additional insights and feedback.

## 2. How can I stay organized while finishing a proof?

Staying organized is crucial when finishing a proof. One way to do this is by keeping track of your progress and any ideas or insights that come to mind. You can use a notebook or a digital document to jot down notes and keep track of your thought process. It is also helpful to use visual aids, such as diagrams or charts, to better understand the problem and its solution.

## 3. What should I do if I get stuck while finishing a proof?

If you get stuck while finishing a proof, take a step back and try to approach the problem from a different angle. You can also try working on a different part of the proof or taking a short break to clear your mind. It is also beneficial to discuss the problem with peers or a mentor for fresh perspectives and ideas.

## 4. How do I know if I have successfully finished a proof?

A proof is considered successful if it follows all the rules and conventions of mathematical reasoning and arrives at a valid and logical conclusion. It should also be clear, concise, and free of any errors. It is essential to double-check your work and have others review it before considering it complete.

## 5. What should I do if I find a mistake in my supposedly finished proof?

Finding a mistake in a seemingly finished proof can be frustrating, but it is essential to remain calm and retrace your steps. Try to identify where the mistake occurred and work towards correcting it. It is also helpful to ask for feedback from others to catch any errors that you might have missed. Remember, mistakes are a natural part of the proof-writing process, and they can be valuable learning opportunities.

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