How Do You Prove A Closure Equals A Union Boundary A?

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SUMMARY

The discussion centers on proving that the closure of a set A equals the union of A and its boundary, denoted as bdA. The closure of A is defined as the set of points x such that for every open rectangle u containing x, the intersection of u and A is non-empty. Participants seek assistance in demonstrating that A closure equals A union bdA and proving that bdA is the intersection of A closure and the closure of the complement of A in R^n.

PREREQUISITES
  • Understanding of topological concepts such as closure and boundary.
  • Familiarity with set notation and operations in R^n.
  • Knowledge of open rectangles and their properties in topology.
  • Basic skills in mathematical proof writing and logic.
NEXT STEPS
  • Study the definitions and properties of closure and boundary in topology.
  • Learn about the intersection of sets and its implications in R^n.
  • Explore examples of closures and boundaries in various topological spaces.
  • Practice writing formal proofs in topology, focusing on closure and boundary relationships.
USEFUL FOR

Mathematics students, particularly those studying topology, and educators looking for clarification on closure and boundary concepts in set theory.

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How do you do this proof? Isn't it already obvious given the definition? I have no idea how to go about writing it down. :confused: If someone could help me with this, I would really appreciate it. Thanks :) Sorry I had to write it in such a messy way.

Define the closure of A as A closure = (x|for every open rectangle u containing x, intersection of u and A not equal to 0)

a) Show that A closure = A union bdA

b) Prove that bdA = intersection of A closure and (R^n-A) closure = bd(R^n-A)
 
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