Proving the Existence of Frontier Points for Sets in ℝ

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SUMMARY

This discussion focuses on proving the existence of frontier points for sets in ℝ, specifically addressing two statements regarding sets E and its complement E^c. The first statement asserts that if a is in E and b is in E^c with a < b, then the intersection of the closed interval [a, b] and the boundary ∂E is non-empty. The second statement claims that the intersection of the open interval (a, b) and ∂E is also non-empty. The participants emphasize the definition of frontier points as limits of sequences from both E and E^c converging to the same limit, highlighting the need for rigorous proof without visual aids.

PREREQUISITES
  • Understanding of set theory and the concept of complements.
  • Familiarity with the definition of frontier points and boundaries in topology.
  • Knowledge of sequences and limits in real analysis.
  • Basic skills in mathematical proof techniques.
NEXT STEPS
  • Study the definition and properties of frontier points in topology.
  • Learn about the convergence of sequences in real analysis.
  • Explore rigorous proof techniques for set properties in ℝ.
  • Investigate examples of sets and their boundaries to solidify understanding.
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in topology and set theory will benefit from this discussion.

ppy
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3.4. Let E[itex]\in[/itex]  R. Prove or disprove the following statements:
(i) if a[itex]\in[/itex]E and b[itex]\in[/itex]E[itex]^{c}[/itex] = ℝ\E and a < b then [a ,b] [itex]\cap[/itex]∂E IS NOT EQUAL TO ∅.
(ii) if a[itex]\in[/itex]E and b[itex]\in[/itex]E[itex]^{c}[/itex] = ℝ\E and a < b then (a ,b) [itex]\cap[/itex]∂E IS NOT EQUAL TO ∅.


I am really stuck I know that the frontier of a set is when a sequence in E and a sequence in E[itex]^{c}[/itex] converge to the same limit.
 
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I am able to show this from drawing a sketch but how can I do it without a drawing?
 

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