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November 12th, 2012, 07:07 AM   #1
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Is the union of infinite disjoint closed sets closed?

Is the union of infinite closed sets (disconnected each other in given topology) closed?
(remark: disconnected each other in given topology, that is, for any two such closed sets A and B there exists two open sets U and V such that U includes A and V includes B, and the intersection of U and V is an emptyset)
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November 12th, 2012, 08:44 AM   #2
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Re: Is the union of infinite disjoint closed sets closed?

Do you mean countably infinite or uncountably infinite? (As it happens the answer is the same in both cases but it's considerably easier to see why in the uncountable case).
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November 12th, 2012, 02:29 PM   #3
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Re: Is the union of infinite disjoint closed sets closed?

Quote:
Originally Posted by 03sqq
Is the union of infinite closed sets (disconnected each other in given topology) closed?
(remark: disconnected each other in given topology, that is, for any two such closed sets A and B there exists two open sets U and V such that U includes A and V includes B, and the intersection of U and V is an emptyset)
Not necessarily. In the usual topology of reals, single points are closed sets. However, the set of all rationals is not a closed set.
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November 12th, 2012, 02:33 PM   #4
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Re: Is the union of infinite disjoint closed sets closed?

If you want a better set meeting your condition, use the set of points {1/n}. These can be separated the way you described, but the union is not a closed set.
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November 13th, 2012, 03:40 AM   #5
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Re: Is the union of infinite disjoint closed sets closed?

Thanks a lot for all replies, especially thanks mathman for the simple and sagacious answer.
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