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cannot have countably infinite dimension.

Proof

(BCT1) The following is a standard proof that a complete pseudometric space is a Baire space.

Let be a countable collection of open dense subsets. We want to show that the intersection is dense.A subset is dense if and only if every nonempty open subset intersects it. Thus to show that the intersection is dense, it suffices to show that any nonempty open subset of has some point in common with all of the .Because is dense, intersects consequently, there exists

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