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PostPosted: Mon, 16 Feb 2009 18:27:13 UTC 
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This question is probably familiar to many, but it is a fun problem.

How many distinct sets are possible using closure and interior, starting from some subset A of some topological space X? In other words, how many distinct subsets can there be in the sequence
\dots,\overline{A^\circ}^\circ,\overline{A^\circ},A^\circ,A,
\overline{A},\overline{A}^\circ,\overline{\overline{A}^\circ},\dots?
For each possibility, give an example of the pair (X,A).


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PostPosted: Mon, 16 Feb 2009 22:20:43 UTC 
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Answer:

Spoiler:
14.


Now can anyone exhibit a set which achieves all of them?

Hint 1:
Spoiler:
Try using \mathbb{R} as your space


Hint 2:
Spoiler:
Be creative with the way you choose the endpoints of your interval(s)


Hint 3: (pretty big)
Spoiler:
Be creative with the connectivity of your set


Hint 4: (BIG)
Spoiler:
Use dense subsets well
.

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PostPosted: Mon, 16 Feb 2009 22:56:44 UTC 
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Shadow --- Taking complement is not allowed here!


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PostPosted: Mon, 16 Feb 2009 23:12:19 UTC 
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:shock:

.
.
.


Well, that's what I get for not reading carefully :roll:. Thanks outermeasure!

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