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Can anybody supply an example of a Baire Space, that is neither locally compact nor metrizable. I would be gratefull also for some references.

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Some such examples were discussed in comments to this question. – Martin Sleziak May 13 '12 at 14:29

As a trivial example, let $X,Y$ be Baire spaces with $X$ not locally compact and $Y$ not metrizable, and consider the disjoint union of $X,Y$.

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The Sorgenfrey line is one example. It is not locally compact nor is it metrizable. Here is a paper showing that it is a Baire space..

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