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Number of topologies on a 3 set

Matthew

Posted: Submitted by Matthew on 4 April 2010 - 1:51pm.

Joined: 2007-11-03
Posts: 12

I'm pretty sure this should be an easy question but I can't get the same answer as everyone else on the internet and I can't see why.

How many topolgies are there on $X=\{1,2,3\}$ ?
I can find 35 but there apparently there should only be 29: Heres my list of the different types of topogies with how many there are of each one.
1. $\{\phi, X\}$ $\times 1$
2. $\{\phi, \{1\}, X\}$ $\times 3$
3. $\{\phi, \{1,2\}, X\}$ $\times 3$
4. $\{\phi, \{1\},\{2,3\},X\}$ $\times 3$
5. $\{\phi, \{1\}, \{1,2\}, X\}$ $\times 6$
6. $\{\phi, \{1\}, \{2\}, \{1,2\}, X\}$ $\times 3$
7. $\{\phi, \{1\}, \{2\}, \{1,3\}, X\}$ $\times 6$
8. $\{\phi, \{1\}, \{1,3\}, \{1,2\}, X\}$ $\times 3$
9. $\{\phi, \{1\}, \{2\}, \{1,2\}, \{2,3\}, X\}$ $\times 6$
10. $P(X)$ $\times 1$
Wheres the mistake?

Callan

Posted: 4 April 2010 - 5:39pm

Joined: 2008-09-30
Posts: 173

I don't think 7 is a topology as the union of $\{1 \}$ and $\{ 2 \}$ is not contained within the topology.

Matthew

Posted: 5 April 2010 - 1:32pm

Joined: 2007-11-03
Posts: 12

Yeah you're right. I actually noticed that as soon as I saw it in latex rather than my scruffy handwriting and thought I'd deleted the thread. Thanks anyways.

johns

Posted: 17 December 2011 - 7:21am

Joined: 2011-12-17
Posts: 15