Finite Supremum and Infinium Solution for Munkres Text Question 20.7

[Fisicks]

I can't seem to find a answer for 20.7 anywhere. Unfourtantly, I do not have the skills to latex the problem out, so I only hope someone looks in the book.

My solution is that the supremum of the set of a_i 's must be finite above and the infinium is finite and greater then zero , and the b_i 's have no restraints.

 

[micromass]

Hi Fisicks!

I can't seem to find a answer for 20.7 anywhere. Unfourtantly, I do not have the skills to latex the problem out, so I only hope someone looks in the book.

My solution is that the supremum of the set of a_i 's must be finite above and the infinium is finite and greater then zero , and the b_i 's have no restraints.

What is that an answer to?? To the continuity of h or to h being a homeomorphism.

For h to be continuous, you are correct: we only need to demand that $\sup{a_i}<+\infty$.

But for h to be homeomorphism, it is also correct, we demand that $\sup{a_i}<+\infty$ and $\inf{a_i}>0$.

Note, the map in this exercise is often called a "diagonal operator". So you can search it by that name

 

[Greg Bernhardt]

Hi there! I understand that you're having trouble finding the answer for 20.7 and you're not able to latex the problem. I would suggest trying to rephrase the problem in a different way and maybe searching for that instead. It's also possible that the answer may not be readily available online and you may need to consult the book or ask a classmate or teacher for help. As for your solution, it sounds like you have a good understanding of the problem and have identified the necessary restraints for the a_i and b_i terms. Keep up the good work!