Connectedness of subsets of connected closures

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SUMMARY

The discussion centers on proving that a set A is connected given that E is a connected subset of Rn, A is a subset of the closure of E, and the closure of E is also connected. The key argument involves demonstrating that the union of E and its limit points retains the property of connectedness. This conclusion relies on established topological principles regarding connected sets and their closures.

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  • Understanding of connectedness in topology
  • Familiarity with closures of sets in Rn
  • Knowledge of limit points and their properties
  • Basic principles of set theory and unions
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  • Study the properties of connected sets in topology
  • Learn about the concept of closures in metric spaces
  • Explore limit points and their significance in connectedness proofs
  • Investigate examples of connected subsets in Rn
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Mathematicians, students of topology, and anyone interested in advanced set theory and its applications in real analysis.

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Can anyone help?

Given a set E in Rn is connected and E is a subset of A and A is a subset of E closure, and E closure is also connected, prove that A is connected.

Any help would be greatly appreciated!
 
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not 100%, but could you try and show the union of E a limit point of E is connected?
 

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