Number of topologies on a 3-point set

[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?

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

Thanks in advance!

 

[micromass]

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

That should help you find all of them.

 

[middleCmusic]

Thanks! I think I got the rest.

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

 

[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.

 

[middleCmusic]

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.

You're right - it also wasn't nearly as confusing as I initially thought. Here's the updated version:

1. The discrete topology

$\tau_{20} = \{X, \emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\} \}$

2. The indiscrete topology

$\tau_1 = \{X, \emptyset\}$

3. A topology with no isolated points (= open singleton sets) and 1 open set consisting of a doubleton

$\tau_5 = \{X, \emptyset, \{a,b\} \}$
$\tau_6 = \{X, \emptyset, \{a,c\} \}$
$\tau_7 = \{X, \emptyset, \{b,c\} \}$

4. A topology with one isolated point and no other [novel] open sets.

$\tau_2 = \{X, \emptyset, \{a\} \}$
$\tau_3 = \{X, \emptyset, \{b\} \}$
$\tau_4 = \{X, \emptyset, \{c\} \}$

5. A topology with one isolated point and the two other points also form an open set.

$\tau_{21} = \{X, \emptyset, \{a\}, \{b,c\} \}$
$\tau_{22} = \{X, \emptyset, \{b\}, \{a,c\} \}$
$\tau_{23} = \{X, \emptyset, \{c\}, \{a,b\} \}$

6. A topology with one isolated point, and the other two points are in its closure, but not in each other's closure

$\tau_{14} = \{X, \emptyset, \{a\}, \{a,b\}, \{a,c\} \}$
$\tau_{15} = \{X, \emptyset, \{b\}, \{a,b\}, \{b,c\} \}$
$\tau_{16} = \{X, \emptyset, \{c\}, \{a,c\}, \{b,c\} \}$

7. A topology with one isolated point, another point is in the closure of it, but not in the closure of the third, while the third is in the closure of both the others.

$\tau_8 = \{X, \emptyset, \{a\}, \{a,b\} \}$
$\tau_9 = \{X, \emptyset, \{a\}, \{a,c\} \}$
$\tau_{10} = \{X, \emptyset, \{b\}, \{a,b\} \}$
$\tau_{11} = \{X, \emptyset, \{b\}, \{b,c\} \}$
$\tau_{12} = \{X, \emptyset, \{c\}, \{a,c\} \}$
$\tau_{13} = \{X, \emptyset, \{c\}, \{b,c\} \}$

8. A topology with 2 isolated points as only non-trivial open sets.

I think this description seems to imply a set of say {X, ø, {a}, {b}} which is wrong, but I think the below is what was intended.

$\tau_{17} = \{X, \emptyset, \{a\}, \{b\}, \{a,b\} \}$
$\tau_{18} = \{X, \emptyset, \{a\}, \{c\}, \{a,c\} \}$
$\tau_{19} = \{X, \emptyset, \{b\}, \{c\}, \{b,c\} \}$

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...

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

These ones seemingly weren't included in Henno's list.

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

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

 

[micromass]

Yes. The description of number 8 is not very accurate.

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

The union of two open sets must be open. Specifically, the union of the singletons above.

 

[middleCmusic]

Ah yes. OK - that clears it up.

 

[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.

 

[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.