Which topological spaces can be written as a countable union of pairwise disjoint clopen subsets?

1.One of them is the Baire space: $\omega ^{\omega} = [0] \cup [1] \cup ....$

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    $\begingroup$ This is a very vague question. What kind of answer would satisfy you? $\endgroup$ – xyzzyz Aug 11 '13 at 2:12
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    $\begingroup$ The discrete topology on a countable set, of course. $\endgroup$ – Thomas Andrews Aug 11 '13 at 2:15
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    $\begingroup$ Pretty much, given any countable sequence of topological spaces, you can define the disjoint union of the spaces. And this classifies all such topological spaces. $\endgroup$ – Thomas Andrews Aug 11 '13 at 2:17
  • $\begingroup$ Well, another example with brief explination OR showing certain zero-dimensional spaces have this property $\endgroup$ – Lo52 Aug 11 '13 at 2:19
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    $\begingroup$ Since you forgot to say in the question that the subsets should be nonempty, the answer is "all spaces". More seriously, if we require the subsets to be nonempty, as you undoubtedly intended, then Thomas Andrews's second comment seems to completely answer the question. One can rephrase it slightly by saying "the disjoint union of any infinite family of nonempty spaces". (If the family is uncountable, you can combine all but countably many of the space into one and then form its disjoint union with the countably many other spaces.) $\endgroup$ – Andreas Blass Aug 11 '13 at 5:01

As noted in the comments, the question is too broad to have any very nice answer that puts the property in a usefully different light: the description in the question is already probably the most natural characterization of such spaces. For what it's worth, here's a trivial characterization that at least looks a little different superficially:

$X$ has a countably infinite partition into clopen sets iff there is a continuous surjection $f:X\to\Bbb N$, where $\Bbb N$ has the discrete topology.


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