[Math] how many topologies on a three element space

general-topology

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!

Best Answer

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...

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