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Let $X$,$Y$ be two finite topological Hausdorff spaces, how can I prove that if the two spaces have the same number of points, they must be homeomorphic?

EDIT: so sorry I missed to add the hausdorff assumption in the body, I thought it was clear from the caption.

Cornman
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conrad
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3 Answers3

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As @David Mitra (in a comment) and @jon.sand (in an answer) have pointed out, this result is true for $T_2$ spaces. In fact, it's even true for $T_1$ spaces. For each positive integer $n$ there is exactly one $T_1$ topology (the discrete topology), up to homeomorphism, on an $n$-element set. This is because for a $T_1$ topological space each singleton set is a closed set, and since every subset of such a space is a finite union of singleton sets, it follows that every subset of such a space is a closed set, and hence every subset of such a space is an open set (i.e. the discrete topology). However, it's not true in a very strong way if we slightly weaken the separation axiom assumption to $T_0.$

A few weeks ago I happened to be looking over the paper cited below, which among other things shows how different the $T_0$ and $T_1$ separation axioms are for finite topological spaces. For example, on a $14$-element set, there are $98,484,324,257,128,207,032,183$ different $T_0$ topologies and $115,617,051,977,054,267,807,460$ different topologies. Or, to put it another way, on a $14$-element set there is exactly one $T_1$ topology and roughly $\frac{1}{6}$ Avogadro's number of $T_0$ topologies!

Marcel Erné and Kurt Stege, Counting finite posets and topologies, Order 8 #3 (1991), 247-265.

Dave L. Renfro
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  • My book has yet not gone through the notion of $T_i$ spaces. This is supposed to be a very elementary question, are you saying, as Jon.sand that this result does not hold? Because i have citet it word by word from my book. – conrad Nov 04 '17 at 15:43
  • After I wrote my answer, I saw "Hausdorff" [= $T_2$] in your title, an assumption that you did not mention in your question, so I assumed you were either trying to prove something that is not true or you simply overlooked an assumption. In trying to be diplomatic, I wrote my answer without assuming which was the case. – Dave L. Renfro Nov 04 '17 at 15:48
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Hint : for a point $x$ in a Hausdorff space, what is the intersection of all open sets containing it ?

If the space is finite, what property does this intersection have ?

Maxime Ramzi
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  • The intersection will consist of just $x$, i dont know what property it will have if the space is finite. Anyhow I dont see at all how I can use this, I am trying to work out explicit continious functions that are inverses. – conrad Nov 04 '17 at 15:36
  • If the space is finite, that intersection is finite, and so is an open set. Therefore ${x}$ is open. Do you think you can go on from here ? – Maxime Ramzi Nov 04 '17 at 15:49
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In general it is not true. Consider discrete topology and anti-discrete topology. But second one is not Hausdorff.

Cornman
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pem
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