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My professor gave us the following question while covering Topological Spaces

Find a list of Topologies $\mathcal{T}_1, \mathcal{T}_2,..., \mathcal{T}_n$ on $X=\{1,2,3\}$ such that for every topology $\mathcal{U}$ on $X$, $(X,\mathcal{U}) \equiv (X,\mathcal{T}_i)$ for some $i$, and $(X,\mathcal{T}_j) \not\equiv (X,\mathcal{T}_k)$ for all $j\neq k$.

In this case '$\equiv$' refers to there being a homeomorphism, topological equivalent, from one topological space to the other.

Now if I understand correctly, a topology is a family of subsets of $X$ satisfying the following condition. (1) $X,\emptyset \in \mathcal{T}$, (2) The union of any family of members of $\mathcal{T}$ is in $\mathcal{T}$, and (3) The intersection of any finite family of members of $\mathcal{T}$ is in $\mathcal{T}$.

Now for two spaces to be topological equivalent, there needs to be a one-to-one correspondence from $X$ to $Y$ for which both $f$ and the inverse function $f^{-1}$ are continuous.

The list of Topologies on $X$ I have, so far, are as follows:

  • $\mathcal{T}_1=\{\emptyset, X\}$ (The Trival Topology)
  • $\mathcal{T}_2=\{\emptyset, X,\{1\}\}$
  • $\mathcal{T}_3=\{\emptyset, X,\{1\}, \{2\}, \{1,2\}\}$
  • $\mathcal{T}_4 =\{\emptyset, X, \{1,2\}\} $
  • $\mathcal{T}_5 =\{\emptyset, X, \{1,2\}, \{2,3\}\} $
  • $\mathcal{T}_6 =\{\emptyset, X, \{1\}, \{1,2\}\} $
  • $\mathcal{T}_7 =\{\emptyset, X, \{3\}, \{1,2\}\} $
  • $\mathcal{T}_8 =\{\emptyset, X, \{1,2\}, \{1,3\}, \{2,3\}\} $
  • $\mathcal{T}_9=\{\emptyset, X,\{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}\}$ (The Discrete Topology)

Not including the Topologies that are equivalent to the ones above, are there any other Topological Spaces on $X$ that I may be missing? Are any of my topologies incorrect?

I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide

Kevin_H
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    By the "power set topology", do you mean the discrete topology? – celtschk Mar 14 '15 at 20:44
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    Maybe a good start is to look at a few topologies to get a feeling of how they work (for example, you could think about the question: What is the smallest topology that contains the set ${1}$? And what is the smallert topology that contains both ${1}$ and ${2}$?). Then think about when topologies are equivalent (hint: a one-to-one correspondence of $X$ to itself is a permutation). – celtschk Mar 14 '15 at 20:52
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    May help http://www.maths.kisogo.com/index.php?title=Topology - "raw" definition - follow Topological space link on the top – Alec Teal Mar 14 '15 at 21:06

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