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### Closed Sets

```
Date: 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/
```
Associated Topics:
College Logic
High School Logic
High School Sets

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