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Dear Clyde Oliver,

I am doing a project in Category Theory where topology is part of my project requirement to study.

I tried to do some questions from some topology textbooks where I do not know how to prove it.

1) SHow that for any open set U of a topological space X, it holds that cl(int(cl(U)))=cl(U). and

2) Using the above show that for any set A, there are at most 14 different sets in the following sequence

i)  A,A^c,A^-c,A^c-c,...,
ii) A,A^-c,A^-c-,A^-c-c,...,

where A^c:= X-A and B^-:= Cl(B).

Can you help me in these two questions. Thanks a lot...

Regards
Raymond

Answer
1) For this question, you'll need to interpret the definitions:

cl(A) = the smallest closed set containing A

int(A) = the largest open set contained in A

So you have U, then you have K=cl(U), a slightly larger set.

Then you have U2=int(K), the largest open set in K=cl(U). U2 must contain U, because U is in K and U2 must be the largest so it is at least as large as U.

Then you have K2=cl(U2). This is the smallest closed set containing U2. Thus K2 contains K, but the assertion is that really, K = K2.

Why? Because K also contains U2, and K2 is the smallest closed set containing U2. If K were not equal to K2, then K would be smaller - not possible.


2) Here, make use of the fact that the interior is the complement of the closure of the complement. I think this is the piece of logic you are missing. Watch how eventually, you wind up "canceling out" some of the operators (c-c becomes int, and then cl-int-cl becomes cl) until you repeat things in the sequence.

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Clyde Oliver

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From "Advanced Math" -- I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks. I have no idea why Topology is not included in "Advanced Math" -- not to mention, there is only one question in this category, and it is decidedly not a topology question.

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I am a PhD educated mathematician working in research at a major university.

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AMS

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Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

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BA mathematics & physics, PhD mathematics from a top 20 US school.

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Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

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In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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