Open cover which has no finite subcover

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SUMMARY

The discussion revolves around providing an example of an open cover in R^n that lacks a finite subcover. The proposed sets are {x ε Q | x < sqrt(3)} and {x ε Q | x > sqrt(3)}. The participants clarify that these sets must be subsets of R^n, open, and collectively cover R^n while demonstrating that no finite subcover can replicate the coverage of the entire open cover. The conclusion confirms that the sets meet the necessary criteria for an open cover without a finite subcover.

PREREQUISITES
  • Understanding of open sets in topology
  • Familiarity with rational numbers and their properties
  • Knowledge of R^n (n-dimensional real space)
  • Concept of finite subcovers in the context of open covers
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  • Study the properties of open sets in topology
  • Learn about the concept of compactness in metric spaces
  • Explore examples of open covers and their finite subcovers
  • Investigate the implications of the Heine-Borel theorem
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Mathematics students, particularly those studying topology, and educators seeking examples of open covers and their properties in R^n.

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Homework Statement



Give an example of an open cover in R^n which has no finite subcover.

Homework Equations

The Attempt at a Solution



{x ε Q | x < sqrt(3)} U {x ε Q | x > sqrt(3) }
 
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None of the elements in that set is an open subset of Rn. None of them are even subsets of Rn!


Or... were you trying to say that you wanted to consider an open cover whose elements are the two sets
{x ε Q | x < sqrt(3)} and {x ε Q | x > sqrt(3) }​
? If so, you need to show four things:

(1) Those two sets are subsets of Rn
(2) Those two sets are open
(3) Those two sets cover Rn
(4) No finite subcover covers the same space as the entire cover does
 
n/m. I've figured this out.
 
Last edited:

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