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Let $X=\{a,b,c\}$ be a three element set, - What are the total number of topologies that can be constructed on this set? - Not every collection of subsets of $X$ is a topology on $X$.Why?

As i've just started topology,i do not have much comfortable setting for this. I need help in understanding how topologies are constructed on an any arbitrary set.

Thanks for help!

MJD
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Styles
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    Did you try enumerating them? The number of topologies on an n-element set is an open problem in general. There is no way around brute forcing them and writing them out. The options are small enough in your case that it is fairly straightforward. – Cameron Williams Jul 23 '16 at 21:42
  • By the way to make ${$ and $}$ render in $\rm\LaTeX$, do { and } inside the $. – Cameron Williams Jul 23 '16 at 21:43
  • @CameronWilliams:yeah,i tried it ,first by considering the number of subsets of X,then by permuting these subsets, but i'm not getting the exact answer – Styles Jul 23 '16 at 21:49
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    @CameronWilliams The article Wikipedia: Finite topological space has some information. In an example they mention that the above $X$ has 29 possible topologies ($X$ has $2^3=8$ subsets, so there are $2^8=256$ collection os subsets, and 29 of those are topologies). Up to homeomorphism there are 9 topological spaces with three points. – Jeppe Stig Nielsen Jul 23 '16 at 23:35
  • Have a look at this http://math.stackexchange.com/questions/32153/alternative-definition-for-topological-spaces?rq=1 –  Jul 24 '16 at 09:14

2 Answers2

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By definition, a topology on a set $X$ is a collection of subsets of $X$ satisfying the following three properties:

  1. The sets $\varnothing$ and $X$ are members of the collection.
  2. The intersection of finitely many members is a member.
  3. The union of an arbitrary subcollection of members is a member.

There are literally only $8$ subsets of $X$ in your case, and you know any topology must already contain $2$ of them. So just try adding in other subsets, making sure that the above three properties are satisfied, and use symmetry to make life easier...

Ashvin Swaminathan
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You can consider the following class of subsets of $X$: $$\mathcal{F}=\lbrace \lbrace a \rbrace, \lbrace b\rbrace, \lbrace c\rbrace , X , \emptyset\rbrace.$$ This class of set isn't a topology on $X$, infact $\lbrace a \rbrace \cup\lbrace b \rbrace= \lbrace a,b\rbrace\not\in \mathcal{F}$.

Vincenzo Zaccaro
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