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

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SUMMARY

A set A that is both open and closed in the real numbers R must be either the empty set or the entire set R itself. This conclusion arises from the definitions of open and closed sets, where an open set does not include its boundary points, while a closed set includes all its boundary points. Therefore, the only sets that can satisfy both conditions simultaneously are the empty set and R.

PREREQUISITES
  • Understanding of open and closed sets in topology
  • Familiarity with boundary points in real analysis
  • Basic knowledge of set theory
  • Concept of subsets in the context of real numbers
NEXT STEPS
  • Study the properties of open and closed sets in topology
  • Explore the concept of boundary points in real analysis
  • Learn about connectedness in topological spaces
  • Investigate the implications of the empty set and full set in set theory
USEFUL FOR

Mathematicians, students of topology, and anyone studying real analysis will benefit from this discussion regarding the properties of sets in the context of real numbers.

seema283k
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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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seema283k said:
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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