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).



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 $\mathbb Q\times\{0\}.$

  • $\begingroup$ Thanks a lot, that's a neat example and it's much simpler than expected! Thanks also for the reference, I didn't know about that book. Now I wish someone could come up with an example not involving $\mathbb{Q}$ (or any other dense first category subset of $\mathbb{R}^n$) :-) $\endgroup$ – Curt F. Jul 26 '13 at 20:06
  • $\begingroup$ It seems that similarly to this example we can embed each Tychonoff space $Y$ as a closed subset into a Baire space $X$. Let $bY$ be an arbitrary compactification of $Y$. Put $X=bY\times [0;1]\setminus (bY\setminus Y)\times\{0\}$. $\endgroup$ – Alex Ravsky Jul 26 '13 at 20:20

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.

  • $\begingroup$ Thank you very much. I forgot to mention explicitly that I'd prefer to have some examples with higher separation properties in order to exclude easy examples such as this one. Nevertheless, I appreciate your help and I hope its okay that I added this requirement to the question. $\endgroup$ – Curt F. Jul 25 '13 at 19:55
  • $\begingroup$ No problem. I would also like to see more natural examples. :) $\endgroup$ – Dune Jul 26 '13 at 8:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.