Examples of closed subspaces of Baire spaces that fail to be Baire?

Question

I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire.

Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category theorem: local compactness or complete metrizability or Čech completeness because all these properties are hereditary with respect to closed subspaces.

An interesting example of a Baire metrizable space that fails to be completely metrizable is given in an answer to What are some motivating examples of exotic metrizable spaces. This example contains $\mathbb Q$ as a closed subspace.

It would be nice to see some further examples.

Added: I would prefer to have examples of high regularity (at least Hausdorff, preferably Tychonoff).

Thanks!

Answer 1

Consider the euclidean topology $\tau$ on the rationals and some point $x$ which is not contained in $\mathbb{Q}$. The space $Q := \mathbb{Q} \cup \{x\}$ with the topology $\tau' := \{ O \cup \{x\} : O \in \tau \} \cup \{ \emptyset \}$ is a Baire space since {x} is dense and contained in every dense subset of $Q$ - so any intersection of dense subsets of $Q$ is again dense. Furthermore $\mathbb{Q}$ is a closed subspace of $Q$ which fails to be Baire - so $(Q,\tau')$ provides an example to your question.

Answer 2

If I right understood the notation of the book "Baire spaces" by R.C Haworth and R.A. McCoy then there is a simple example of a Baire space $\mathbb{ R^2\setminus ((R\setminus Q)\times \{0\}})$ with closed non-Baire subset $\mathbb Q\times\{0\}.$