Baire property is open subspace-closed

Verbal statement
Every open subset of a Baire space is itself a Baire space, under the subspace topology.

Baire space
A topological space is termed a Baire space if an intersection of countably many open dense subsets is dense.

Facts used

 * 1) uses::Open subset of open subspace is open

Proof
Given: A Baire space $$X$$, an open subset $$A$$. A countable family of open dense subsets, $$U_n, n \in \mathbb{N}$$ of $$A$$

To prove: The intersection $$T = \bigcap_{n \in \mathbb{N}} U_n$$ is dense in $$A$$

Proof:

Textbook references

 * , Page 297, Lemma 48.4, Chapter 4, Section 48