Closed SetsDate: 02/27/99 at 07:37:29 From: Michael Subject: Closed Sets Problem We showed in class that a set is closed if and only if it contains all of its limit points. I want to know how to use this property to show that a union of finite closed sets and the intersection of any number of closed sets are closed. Date: 02/27/99 at 07:50:43 From: Doctor Allan Subject: Re: Closed sets problem Assume the union of closed sets C_1, C_2,..., C_n is not closed. This means that there is a limit point, x, outside the union. All limit points of the union must belong to one of the C_i's (why?), so x must belong to one of the C_i's. Does it belong to C_1? No, because if it did, it would belong to the union as well. In addition, it does not belong to any of the other C_i's, yielding a contradiction. Therefore the union must be closed. Similar reasoning can be used for your question on intersection. Sincerely, - Doctor Allan, The Math Forum http://mathforum.org/dr.math/ |
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