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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?
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?
Several examples of this kind are mentioned at standard places where to look for counterexamples in general topology. Wikipedia article on totally disconnected spaces mentions Erdős space. The same space is also mentioned as Example 6.2.19 in Ryszard Engelking: General Topology, Heldermann Verlag...