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Is a single point in R compact?

  1. Jun 27, 2009 #1
    Is a single point in R compact?

    It seems obvious since every open cover of a single point in R can clearly have a finite subcover.

    However, I have a little uncertainty (i.e possible convention that says otherwise?) so just wanted to check before using it in a proof.
    thanks
     
  2. jcsd
  3. Jun 27, 2009 #2
    A subset of R is compact if it is closed and bounded (Heine Borel). A set consisting of a single point is certainly bounded and closed and therefore compact
     
  4. Jun 27, 2009 #3

    Office_Shredder

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    That's it. No more work needed

    I would probably fail anyone using this argument on principle
     
  5. Jun 27, 2009 #4

    HallsofIvy

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    Technical point: it makes no sense to talk about a point being compact. What you mean is that a set containing a single point (a "singleton" set) is compact. That's true in any topology, not just R or even just in a metric space. Given any open cover for {a}, there exist at least one set in the cover that contains a and that set alone is a "finite subcover".
     
  6. Jun 27, 2009 #5
    And I would say you would be in error to do so.

    What would you say to this:

    Let [tex]p[/tex] be the point. We know that there is some compact space [tex]K[/tex]. The map defined by [tex]f(x) = p[/tex] for all [tex]x \in K[/tex] is continuous. The continuous image of a compact set is compact. Therefore [tex]\{p\}[/tex] is compact. QED
     
  7. Jun 27, 2009 #6

    Office_Shredder

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    I was mostly being humorous. However, I know a lot of teachers that specifically don't want people to use Heine-Borel to prove the compactness of sets because it misses the point (which is to demonstrate your knowledge of what compactness means).
     
  8. Jun 27, 2009 #7

    HallsofIvy

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    Oh, surely you can find an even more complicated proof than that!
     
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