# Number of topologies on a 3-point set

1. Feb 23, 2013

### middleCmusic

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?

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

2. Feb 23, 2013

### micromass

$$\{\emptyset,X,\{a\},\{b,c\}\}$$
$$\{\emptyset,X,\{a\},\{b\},\{a,b\},\{b,c\}\}$$

That should help you find all of them.

3. Feb 23, 2013

### middleCmusic

Thanks! I think I got the rest.

$\tau_{21} = \{X, \emptyset, \{a\}, \{b,c\} \}$
$\tau_{22} = \{X, \emptyset, \{b\}, \{a,c\} \}$
$\tau_{23} = \{X, \emptyset, \{c\}, \{a,b\} \}$
$\tau_{24} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\}, \{b,c\} \}$
$\tau_{25} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\} \}$
$\tau_{26} = \{X, \emptyset, \{a\}, \{c\}, \{a,c\}, \{b,c\} \}$
$\tau_{27} = \{X, \emptyset, \{a\}, \{c\}, \{a,b\}, \{a,c\} \}$
$\tau_{28} = \{X, \emptyset, \{b\}, \{c\}, \{a,b\}, \{b,c\} \}$
$\tau_{29} = \{X, \emptyset, \{b\}, \{c\}, \{a,c\}, \{b,c\} \}$

4. Feb 23, 2013

### micromass

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.

5. Feb 23, 2013

### middleCmusic

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...

$\tau_{30} = \{X, \emptyset, \{a\}, \{b\}, \{a,c\} \}$
$\tau_{31} = \{X, \emptyset, \{a\}, \{b\}, \{b,c\} \}$
$\tau_{32} = \{X, \emptyset, \{a\}, \{c\}, \{a,b\} \}$
$\tau_{33} = \{X, \emptyset, \{a\}, \{c\}, \{b,c\} \}$
$\tau_{34} = \{X, \emptyset, \{b\}, \{c\}, \{a,b\} \}$
$\tau_{35} = \{X, \emptyset, \{b\}, \{c\}, \{a,c\} \}$

Uh oh... now I'm getting 35... So either the site is wrong or something is repeated or something isn't a topology.

6. Feb 23, 2013

### micromass

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.

7. Feb 23, 2013

### middleCmusic

Ah yes. OK - that clears it up.

8. Feb 27, 2013

### WWGD

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.

9. Feb 28, 2013

### Bacle2

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.