Basic Topology Proof: y in E Closure if E is Closed

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SUMMARY

The discussion centers on the proof that if \( y = \sup E \) for a nonempty set \( E \) of real numbers that is bounded above, then \( y \in \text{closure}(E) \) if \( E \) is closed. The closure of \( E \) is defined as \( \text{closure}(E) = E' \cup E \), where \( E' \) represents the set of limit points of \( E \). The proof correctly identifies that if \( E \) is closed, then \( E' \subset E \), leading to the conclusion that \( \text{closure}(E) = E \). Thus, \( y \in \text{closure}(E) \) is validated.

PREREQUISITES
  • Understanding of real number sets and supremum.
  • Familiarity with the concept of closure in topology.
  • Knowledge of limit points and their relationship to closed sets.
  • Basic proof techniques in mathematical analysis.
NEXT STEPS
  • Study the properties of closed sets in topology.
  • Learn about limit points and their significance in real analysis.
  • Explore the concept of supremum and infimum in ordered sets.
  • Review formal proof techniques in mathematical logic.
USEFUL FOR

Mathematics students, particularly those studying real analysis and topology, as well as educators looking for examples of proofs involving closure and limit points.

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Homework Statement



Let E be a nonempty set of real numbers which is bounded above. Let y=sup E. Then y [itex]\in[/itex] E closure. Hence y [itex]\in[/itex] E if E is closed.

Homework Equations



E closure = E' [itex]\cup[/itex] E where E' is the set of all limit points of E.

The Attempt at a Solution



By the definition of closure, y is either in E' or E (maybe in both). If E is closed, E' [itex]\subset[/itex] E and we know that the union of a set and its subset is the set itself. Therefore E closure = E.

Is this a valid proof?
 
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Shouldn't you be proving that if y=sup(E), then y ∈ E closure?
 

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