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Dear Sir,

I am sorry for being imprudent. I think you are right that I should try to comprehend it even though my priority is my category theory, that is what learning is all about. Thanks for your link to one of my questions. If you do not mind, I would like to seek for your advice by showing you my coarse proofing to the second question which I posted to you yesterday.

Let X be a set and A subset= X.

(i) Show that Tau:={U subset=X such that A subset=U} is a topology on X.

(ii) Describe the closure of a subset B with respect to the topology Tau in (i).

This is how I do it. Please advise me if my understanding is incorrect.

To prove the above a topology, we need to show the 3 axiom defined topology

1) Let Ui belong tau for all i belong to tau. SInce A subset U implies that A subset UUi. Thus, Ui belong to tau.

2) Again we let Ui belong to tau for all 1<=i<=n. As A subset U thus A subset intersection Ui for all 1<=i<=n. Hence, intersection Ui belong to tau.

3) As A subset U, thus we have X belong to tau. But I do not know how to show empty set as the definiton doesnt include empty set. Can you give me some advice on this.

ii) Given the closure B is a subset of this topology. By definition, intersection of all closed sets k such that B subset k. We know tau is a topology thus every member of tau will be open set. Since K is closed then complement of k is open which A subset of K^c. This means that A intersect k^c is empty set. Hence, Cl(B)= B.

Your explanation of part i (that is, items 1 through 3) is fine, but before you start proving 1 through 3 you should remind us that A is a fixed set maybe.

For part 2, you seem to be assuming that A is a subset of K^c (and what is K? is it K=cl(B)?). In any case, your answer seems to be wrong unless B and A are disjoint.

Consider X = {1,2,3} and A={1}. Then Tau = { {1}, {1,2}, {1,3}, {1,2,3} }

What if B={2} or B={1}? Well, the first case {2} is closed because {1,3} is open.

But that is not true if B={1}. You should reconsider a bit about whether there are "cases" you need to check. It seems like if you assume B and A don't intersect, you are right.

If B and A intersect, what happens?

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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