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I'm self-teaching out of Morris'sTopology Without Tearsand 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!

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# Number of topologies on a 3-point set

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