Any $\ T_0$ space that has a base consisting of closed (hence clopen) sets is totally disconnected. Does a totally disconnected space necessarily have a base consisting of closed sets?
Totally disconnected implies base of closed sets?
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$\begingroup$
general-topology
examples-counterexamples
connectedness
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2I think the converse is true if you add in "locally compact Hausdorff". – 2012-05-07
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0Willard's *General Topology* has one counterexample in exercise 29B. See [here](http://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA214#v=onepage&q&f=false). There are others; c.f. *Counterexamples in Topology* Steen and Seebach. – 2012-05-07
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0For the case that $X$ is compact, the result mentioned by Dylan is shown in this interesting blog post: [Thoughts about connectedness (Totally disconnected spaces)](http://drexel28.wordpress.com/2010/03/28/thoughts-about-connectedness-totally-disconnected-spaces/). – 2012-06-01