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If a set A is both open and closed then it is R(set of real numbers)

by seema283k
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seema283k
#1
Sep10-09, 10:20 AM
P: 4
if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way
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#2
Sep10-09, 02:36 PM
P: 810
Quote Quote by seema283k View Post
if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way
First, are you talking about an open-and-closed subset of R? If you are, then the empty set is also open-and-closed in R, so you have to specify that A is nonempty.

My approach for this proof would be to consider this.

An open set contains none of its boundary points.
A closed set contains all its boundary points.

The only way for this to be possible is for A to have NO boundary points at all. Show how {} and R are the only two sets that have this property.


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