I wonder if there exists a locally connected metric space of first category.
I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I specifically wanted to avoid mixing locally connected with completely metrizable because it would then be locally arcwise connected and the claim of my theorem would be trivialized.
I took some pains to demonstrate that any Bernstein subset of the plane is a connected locally connected hereditarily Baire space that is not completely metrizable just to show that my theorem is not about nothing. However, separability of the space will also make my theorem useless, so I have not managed to demonstrate that my theorem increases the class of spaces for which the claim holds.
But it is not easy to even find a locally connected metric space of first category. No such example is given in the book Counterexamples in Topology.