Is the Finite Cartesian Product of Simply Connected Sets Also Simply Connected?

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In summary, the question is about the connectedness of the Cartesian product of simply connected sets. It is a natural extension of the theorem that the finite Cartesian product of connected sets is connected, but it requires some knowledge of algebraic topology. However, it seems easy to prove that the product of two simply connected spaces is also simply connected by showing that path connectedness and triviality of the fundamental group are preserved.
  • #1
jimisrv
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Hi,

Easy question about connectedness. I have from Munkres that the finite Cartesian product of connected sets is connected. How about for simply connected sets? This seems like a natural extension of the theorem. Would you say I need to offer a proof? Simply connectedness seems to require some algebraic topology that I am not really familiar with.

Thanks,
Mike
 
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  • #2
If you haven't seen the notion of simple connectedness, and are not expected to have seen it in the context in which your problem arises, then it probably means that there is another way to go.

But the fact that the product of two simply connected space is again simply connected seems easy to prove to me. You must show that path connectedness and triviality of the fundamental group is preserved by taking the product.
 

1. What is a simply connected question?

A simply connected question is a type of question that can be answered with a simple yes or no response. It does not require any further explanation or clarification.

2. How is a simply connected question different from other types of questions?

Unlike other types of questions that may require more detailed answers or multiple responses, a simply connected question only requires a single, straightforward answer.

3. What is the purpose of using simply connected questions in research or experiments?

Simply connected questions are often used in research or experiments as they can help gather specific and concise data. They also allow for easier analysis and interpretation of results.

4. Can a simply connected question lead to biased or inaccurate results?

Yes, if a simply connected question is poorly worded or biased, it can lead to inaccurate or biased results. It is important to carefully construct simply connected questions in order to obtain reliable data.

5. Are there any limitations to using simply connected questions?

Yes, simply connected questions may not be suitable for gathering in-depth or complex information. They are best used for obtaining basic and straightforward data.

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