Prove that countable intersections of closed subset of R^d are closed

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SUMMARY

The discussion focuses on proving that countable intersections of closed subsets of R^d are closed. It emphasizes the importance of considering convergent sequences within these intersections, asserting that closed sets must include the limits of their convergent sequences. Additionally, the conversation suggests exploring the properties of countable unions of open subsets of R^d, reinforcing the foundational concepts of topology in real analysis.

PREREQUISITES
  • Understanding of closed sets in topology
  • Familiarity with convergent sequences in R^d
  • Knowledge of open sets and their properties
  • Basic concepts of real analysis
NEXT STEPS
  • Study the properties of countable unions of open subsets in R^d
  • Explore the concept of convergence in metric spaces
  • Investigate the relationship between closed sets and limit points
  • Learn about the topology of R^d and its implications in analysis
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Mathematicians, students of real analysis, and anyone interested in the properties of topological spaces and their intersections.

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Prove that countable intersections of closed subset of R^d are closed
 
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Try proving that countable unions of open subsets of R^d are open.
 
If the intersection is not empty (what happens if it is?), consider a convergent sequence x(n) in it; what must happen to the limit? (Remember that closed sets must contain the limits of their convergent sequences).
 

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