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Closed or open question about the Extended Real line

  1. Oct 8, 2012 #1
    This is just a quick question about sets that include plus or minus infinity on the extended real line. I am wondering about this in regards to measure in analysis so specifically, is [-∞,a) open or closed? I hadn't seen the extended reals before this class and we really didn't spend anytime discussing them. I want to say it is open since it certainly does not have a boundary point on the a) side and it seems like there isn't anything to approach the boundary on the [-∞. Moreover, it's complement [a,∞] looks suspiciously like a closed set.... Any verification on these musings? Thanks
     
  2. jcsd
  3. Oct 9, 2012 #2
    This link should help. Go to Section 6 - Extended Reals, see the 2nd paragraph.

    http://math.rice.edu/~semmes/math443.pdf

    Also, keep in mind that a set may be neither open nor closed. It might be interesting to find a subset of the extended reals that is neither open nor closed to see how the definitions/requirements given may fail.
     
    Last edited: Oct 9, 2012
  4. Oct 9, 2012 #3
    So then it's not open since it isn't open in respect to both τ+ and τ-?
     
  5. Oct 9, 2012 #4

    mfb

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    How do you come to that conclusion?
    It is wrong, unless you use some non-standard topology.

    Or both at the same time.
     
  6. Oct 9, 2012 #5
    Ah, well now I think it is open. I misunderstood the bit about τ+- in the link. Am I wrong?
     
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