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Topology/Regular Space

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Question
Dear Clyde Oliver,

I have a problem with the following question.

1)Let X be a regular space. Show that for each closed set A, A is the intersection of all open sets containing A. A set which is the intersection of all open sets containing it is called a saturated set. Is the converse of this conclusion true?

I didnt have any notion to start with it. Can you help me?

Thanks in advance.

Answer
Because this appears to be a homework question, it would be inappropriate to give a full answer. Instead, I will try to provide some guidance:

The goal is to prove "closed" ↠ "saturated."

The definitions:

A is closed if X\A, the complement of A in X, is open in the topology.

A is saturated if A = ∪ U, the union of all open sets U such that A⊆U.

X is regular if for any nonempty closed set K, and x not in K, there are open sets U and V that do not intersect, where U is contained in K and V contains x.


Now, what you need to do is to prove that this intersection is exactly K. Say that the intersection of all open sets containing K is some set K'. You wish to prove K'=K.

Then assume otherwise -- assume there is a point k' inside of K' that is not in K. Can you find an open set containing K but not containing k'? If so, then k' should not be in K' after all, leading to a contradiction. Just apply the definition of regular space to obtain this result.

I won't give more details -- as a homework problem, you need to work out the details on your own. As for the converse, consider A = any open set. Is A saturated? Is A likely to be closed?

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Clyde Oliver

Expertise

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.

Experience

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

Organizations
AMS

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

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

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

Past/Present Clients
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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