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Number of topologies on a 3-point set

  1. Feb 23, 2013 #1
    Hey guys,

    I'm self-teaching out of Morris's Topology Without Tears and I'm trying to figure out all of the topologies of a 3-point set {a,b,c}. I came up with 20, but when I checked online, this site said there were 29: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2000;task=show_msg;msg=0041.0001

    I didn't learn anything by going through the description (despite trying) because they didn't give an explicit list of them, and I couldn't figure out where mine wasn't matching up. Here's what I have - can anyone name a topology that I've missed?

    [itex]\tau_1 = \{X, \emptyset\}[/itex]
    [itex]\tau_2 = \{X, \emptyset, \{a\} \}[/itex]
    [itex]\tau_3 = \{X, \emptyset, \{b\} \}[/itex]
    [itex]\tau_4 = \{X, \emptyset, \{c\} \}[/itex]
    [itex]\tau_5 = \{X, \emptyset, \{a,b\} \}[/itex]
    [itex]\tau_6 = \{X, \emptyset, \{a,c\} \}[/itex]
    [itex]\tau_7 = \{X, \emptyset, \{b,c\} \}[/itex]
    [itex]\tau_8 = \{X, \emptyset, \{a\}, \{a,b\} \}[/itex]
    [itex]\tau_9 = \{X, \emptyset, \{a\}, \{a,c\} \}[/itex]
    [itex]\tau_{10} = \{X, \emptyset, \{b\}, \{a,b\} \}[/itex]
    [itex]\tau_{11} = \{X, \emptyset, \{b\}, \{b,c\} \}[/itex]
    [itex]\tau_{12} = \{X, \emptyset, \{c\}, \{a,c\} \}[/itex]
    [itex]\tau_{13} = \{X, \emptyset, \{c\}, \{b,c\} \}[/itex]
    [itex]\tau_{14} = \{X, \emptyset, \{a\}, \{a,b\}, \{a,c\} \}[/itex]
    [itex]\tau_{15} = \{X, \emptyset, \{b\}, \{a,b\}, \{b,c\} \}[/itex]
    [itex]\tau_{16} = \{X, \emptyset, \{c\}, \{a,c\}, \{b,c\} \}[/itex]
    [itex]\tau_{17} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\} \}[/itex]
    [itex]\tau_{18} = \{X, \emptyset, \{a\}, \{c\}, \{a,c\} \}[/itex]
    [itex]\tau_{19} = \{X, \emptyset, \{b\}, \{c\}, \{b,c\} \}[/itex]
    [itex]\tau_{20} = \{X, \emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\} \}[/itex]

    Thanks in advance!
  2. jcsd
  3. Feb 23, 2013 #2

    That should help you find all of them.
  4. Feb 23, 2013 #3
    Thanks! I think I got the rest.

    [itex]\tau_{21} = \{X, \emptyset, \{a\}, \{b,c\} \}[/itex]
    [itex]\tau_{22} = \{X, \emptyset, \{b\}, \{a,c\} \}[/itex]
    [itex]\tau_{23} = \{X, \emptyset, \{c\}, \{a,b\} \}[/itex]
    [itex]\tau_{24} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\}, \{b,c\} \}[/itex]
    [itex]\tau_{25} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\} \}[/itex]
    [itex]\tau_{26} = \{X, \emptyset, \{a\}, \{c\}, \{a,c\}, \{b,c\} \}[/itex]
    [itex]\tau_{27} = \{X, \emptyset, \{a\}, \{c\}, \{a,b\}, \{a,c\} \}[/itex]
    [itex]\tau_{28} = \{X, \emptyset, \{b\}, \{c\}, \{a,b\}, \{b,c\} \}[/itex]
    [itex]\tau_{29} = \{X, \emptyset, \{b\}, \{c\}, \{a,c\}, \{b,c\} \}[/itex]
  5. Feb 23, 2013 #4
    It might be a good exercise to match each of these 29 topologies up with the 9 categories in your link. I think it's quite important for you to understand the descriptions the way they presented them.
  6. Feb 23, 2013 #5
    You're right - it also wasn't nearly as confusing as I initially thought. Here's the updated version:

    9. A topology with 2 isolated points, and the third one is in the closure
    of one of them.

    I didn't have these ones before...

    [itex]\tau_{30} = \{X, \emptyset, \{a\}, \{b\}, \{a,c\} \}[/itex]
    [itex]\tau_{31} = \{X, \emptyset, \{a\}, \{b\}, \{b,c\} \}[/itex]
    [itex]\tau_{32} = \{X, \emptyset, \{a\}, \{c\}, \{a,b\} \}[/itex]
    [itex]\tau_{33} = \{X, \emptyset, \{a\}, \{c\}, \{b,c\} \}[/itex]
    [itex]\tau_{34} = \{X, \emptyset, \{b\}, \{c\}, \{a,b\} \}[/itex]
    [itex]\tau_{35} = \{X, \emptyset, \{b\}, \{c\}, \{a,c\} \}[/itex]

    Uh oh... now I'm getting 35... So either the site is wrong or something is repeated or something isn't a topology.
  7. Feb 23, 2013 #6
    Yes. The description of number 8 is not very accurate.

    The union of two open sets must be open. Specifically, the union of the singletons above.
  8. Feb 23, 2013 #7
    Ah yes. OK - that clears it up.
  9. Feb 27, 2013 #8


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    Science Advisor
    Gold Member

    This should also be an interesting exercise, but for general spaces: find the largest and
    smallest non-trivial ( meaning neither discrete nor indiscrete) topologies on any set.
  10. Feb 28, 2013 #9


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    Science Advisor

    Wow, the (a) smallest one is not so hard but (a) largest one I can do it but only
    using ultrafilters and some kind-of-heavy machinery; I can't see now how to find
    one in a more elementary way.
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