A space $X$ is called a Baire space if every countable intersection of open dense sets is dense. By the Baire category theorem, every complete metric space is Baire and every locally compact Hausdorff space is Baire.
The Sorgenfrey line is an example of a Baire space (shown here) that is not metrizable and not locally compact.
The Sorgenfrey plane and the Niemytzki/Moore plane are also not metrizable and not locally compact, and are not even normal.
For reference, I'd like a proof that the Sorgenfrey plane and the Niemytzki plane are Baire spaces. Sketch of proof is fine.
Best Answer
For the Moore plane: If a space has an dense subspace which is a Baire space, then the space itself is Baire. The open half-plane is dense in the Moore plane and is Baire.