Whats the difference between a closed set vs a set thats NOT open?

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SUMMARY

The discussion clarifies that a set being "not open" does not equate to it being "closed." It highlights that most subsets of real numbers, such as the natural numbers (N) and the interval [0,1), are neither open nor closed. The concept of "clopen" sets, which are both open and closed, is introduced, emphasizing that in topology, a closed set is defined as having an open set as its complement. The standard topology on real numbers is referenced to illustrate these points.

PREREQUISITES
  • Understanding of basic topology concepts, including open and closed sets.
  • Familiarity with the standard topology on real numbers.
  • Knowledge of interior points and their significance in set classification.
  • Comprehension of clopen sets and their properties.
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  • Study the properties of open and closed sets in topology.
  • Explore the concept of clopen sets in various topological spaces.
  • Learn about the implications of continuity in functions between topological spaces.
  • Investigate the definitions and examples of interior and exterior points in set theory.
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Students and professionals in mathematics, particularly those focusing on topology, set theory, and real analysis, will benefit from this discussion.

michonamona
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Let f: D->R be continuous. If D is not open, then f(D) is not open.

Why can they not replace 'not open' with closed?

Thank you

M
 
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Not open does not imply closed. Most subsets of the real numbers are neither open nor closed (e.g. N, or [0,1), etc.). On the other hand, the set of all real numbers is both open and closed.

If a set A is not open then there exists a point x in A such that x is not an interior point of A. For instance, 0 is not interior to the interval [0,1), so its not open. Yet the interval is still "half-open", so it can't be closed.
 
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blkqi's answers is the one relevant to your question, but going more generally, into the topological definition, a topology is a set S plus a collection of special subsets T which is closed under finite intersections and arbitrary unions, and such that both S and the null set are in T. Then a closed set is one which has an open set as complement. Thus both S and the null set are "clopen", both closed and open. There again you see that "not open" and "closed" are not synonyms.
Taking the standard topology on the real numbers, this reduces to blkqi's answer.
 

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