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Open cover which has no finite subcover

  1. Oct 10, 2009 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations


    3. The attempt at a solution

    {x ε Q | x < sqrt(3)} U {x ε Q | x > sqrt(3) }
     
  2. jcsd
  3. Oct 10, 2009 #2

    Hurkyl

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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
     
  4. Oct 10, 2009 #3
    n/m. I've figured this out.
     
    Last edited: Oct 10, 2009
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