# A locally connected metric space of first category

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.

• I'm surprised that Bernstein sets are so nice (connected, locally connected and hereditarily Baire). It's clear they're not completely metrisable (or they'd be $G_\delta$ and measurable), but could you show the other properties? – Henno Brandsma Jun 11 '14 at 17:23
• That a Bernstein set is hereditarily Baire is proved on page 30 in apronus.com/math/MRWojcikPhD.htm. – user156495 Jun 11 '14 at 19:33

## 1 Answer

Consider $$X=\mathbb{Q}\times\mathbb{R}\cup\mathbb{R}\times\mathbb{Q}\subset\mathbb{R}^2$$. Note that $$X$$ is locally connected: you can travel between any two points in an open ball of $$X$$ by moving along horizontal and vertical lines. But $$X$$ is the union of the countably many lines $$\{q\}\times\mathbb{R}$$ and $$\mathbb{R}\times\{q\}$$ for $$q\in\mathbb{R}$$ which are closed and have empty interior, so $$X$$ is first category.

• Thanks. This answers the question as stated but I was thinking of avoiding arcs. – user156495 May 11 '19 at 14:59