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Math Help - Size of Set of Topologies on an Infinite Set

  1. #1
    Super Member redsoxfan325's Avatar
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    Size of Set of Topologies on an Infinite Set

    Let X be an infinite set and let T=\{\mathcal{T}~|~\mathcal{T}~\mbox{is~a~topology~  on~} X\}

    How big is |T|?

    What if we restrict T to be the set of all the topologies on X up to homeomorphism?

    It's pretty clear that an upper bound (in either case) is |2^{2^X}|. In the first case, |X| is definitely a lower bound, because for each x\in X, we can define \mathcal{T}_x=\{\emptyset,\{x\},X\}. In the second case, it's got to be infinite, because you can take a countable subset \{x_1,x_2,...\} and define the topologies \mathcal{T}_1=\{\emptyset,\{x_1\},X\} and \mathcal{T}_n=\mathcal{T}_{n-1}\cup\{U\cup\{x_n\}~|~U\in\mathcal{T}_{n-1}\}.

    Beyond this, I'm not too sure.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    Let X be an infinite set and let T=\{\mathcal{T}~|~\mathcal{T}~\mbox{is~a~topology~  on~} X\}

    How big is |T|?

    What if we restrict T to be the set of all the topologies on X up to homeomorphism?

    It's pretty clear that an upper bound (in either case) is |2^{2^X}|. In the first case, |X| is definitely a lower bound, because for each x\in X, we can define \mathcal{T}_x=\{\emptyset,\{x\},X\}. In the second case, it's got to be infinite, because you can take a countable subset \{x_1,x_2,...\} and define the topologies \mathcal{T}_1=\{\emptyset,\{x_1\},X\} and \mathcal{T}_n=\mathcal{T}_{n-1}\cup\{U\cup\{x_n\}~|~U\in\mathcal{T}_{n-1}\}.

    Beyond this, I'm not too sure.
    This is an interesting question I myself have asked. It turns out that there is no known formula for the number of topologies on a set with finite cardinality, and my guess (though this could be wrong) is that there is not a given formula for infinite sets. In fact, a lot of interesting things about the number of topologies on a finite set (like the number of topologies is the same as the number of preorders, and the number of Kolomogorov topologies is the number of partial orderings) can't be said in general for infinite sets. I'll give this more though and get back to you.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    What might be of interest to you also is the lemma in the last post of this post. This shows that with certain conditions topological spaces can only have certain amounts of cardinality. So, for example if X is a T_1 separable space then \text{card }X\leqslant 2^{\mathfrak{c}} where \mathfrak{c}=2^{\aleph_0}.
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