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Math Help - Totally Disconnected Problem

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    Super Member Aryth's Avatar
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    Totally Disconnected Problem

    I have a problem that is probably relatively easy, I'm just a little lost since I've just started our chapter on connectedness. Here's the problem:

    If (X,\Omega) is a totally disconnected, compact space, prove that (X,\Omega) has a clopen basis.

    Our professor gave us a hint: For each x\in X, let U be an open set containing x and work with the complement Y = X - U to find a clopen V such that x\in V\subseteq U.

    I guess my problem is that I'm not seeing what a clopen set in this context looks like, and what compactness has to do with this. Any help to get me started is appreciated.
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    Re: Totally Disconnected Problem

    Compactness comes a little later. Note that there are two definitions of compactness that might benefit you in this case. Given a collection of closed sets with the finite intersection property, their intersection is nonempty (this is equivalent to the normal definition of compactness).

    Anyway, you know that the complement of an open set is closed, and a closed subset of a compact space is compact. So, Y is compact. Additionally, X is totally disconnected, so any cover of Y by open sets can be reduced to a finite subcover. Total disconnectedness means that every nontrivial subset of X is not connected. That will be useful to finding clopen subsets. The intersection of closed sets remains closed, but only the intersection of finitely many open sets remains open. So, be careful when intersecting clopen sets.

    Disclaimer: It has been years since I have done this problem, so I do not recall off the top of my head all the steps I took to solve it. I am not sure if the alternate definition of compactness was useful in this proof. Also, I seem to recall a picture that looked like an atlas. Each clopen subset of X is a totally disconnected compact space. So, I cut up X into "states", then I cut up each state into "counties", etc. (At least, that's how the picture looked to me).
    Last edited by SlipEternal; November 6th 2013 at 06:55 PM.
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    Super Member Aryth's Avatar
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    Re: Totally Disconnected Problem

    Quote Originally Posted by SlipEternal View Post
    Compactness comes a little later. Note that there are two definitions of compactness that might benefit you in this case. Given a collection of closed sets with the finite intersection property, their intersection is nonempty (this is equivalent to the normal definition of compactness).

    Anyway, you know that the complement of an open set is closed, and a closed subset of a compact space is compact. So, Y is compact. Additionally, X is totally disconnected, so any cover of Y by open sets can be reduced to a finite subcover. Total disconnectedness means that every nontrivial subset of X is not connected. That will be useful to finding clopen subsets. The intersection of closed sets remains closed, but only the intersection of finitely many open sets remains open. So, be careful when intersecting clopen sets.

    Disclaimer: It has been years since I have done this problem, so I do not recall off the top of my head all the steps I took to solve it. I am not sure if the alternate definition of compactness was useful in this proof. Also, I seem to recall a picture that looked like an atlas. Each clopen subset of X is a totally disconnected compact space. So, I cut up X into "states", then I cut up each state into "counties", etc. (At least, that's how the picture looked to me).
    I appreciate it, your descriptions help a lot. I can at least see where I'm supposed to go with it, hopefully the trials and errors will lead me to the solution.
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