PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Open subspaces and residual subspaces of Baire spaces are Baire

Statement

Every open subspace of a Baire space is Baire. Every residual subspace of a Baire space, with its subspace topology, is Baire.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

For a topological space $X$ , the following are equivalent: every countable intersection of dense open sets is dense; every countable union of closed sets with empty interior has empty interior; no nonempty open subset is meagre in $X$ ; and every residual subset meets every nonempty open set. The equivalence includes the empty space. (Equivalent forms of the Baire property).

[F2]

A set is residual when its complement is contained in the union of one sequence of nowhere dense sets (Nowhere dense, meagre, residual, and comeagre subsets of a topological space).

[F3]

Let $(X, T)$ be a topological space (def-topological-space) and let $S \subseteq X$ . The subspace topology (also relative topology) on $S$ is $T_{S} : = {U \cap S : U \in T},$ the family of traces on $S$ of the open sets of $X$ . The pair $(S, T_{S})$ is a subspace of $X$ . A subset of $S$ that lies in $T_{S}$ is said to be open in $S$ , and relatively open where the ambient space needs emphasis. (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Proof

technique · direct

1.1F1F3given

Let $O$ be open in a Baire space $X$ , and let $(G_{n})$ be dense open subsets of $O$ . Put $F_{n} = O ∖ G_{n}$ and $C_{n} = {\overline{F_{n}}}^{X}$ . Each $C_{n}$ is nowhere dense in $X$ : if a nonempty $X$ -open $V$ lay in $C_{n}$ , then $V$ would meet $O$ (since $C_{n} \subseteq \overline{O}$ ), while the nonempty relatively open $V \cap O$ would lie in $C_{n} \cap O = F_{n}$ , contradicting density of $G_{n}$ in $O$ .

1.2F1F2given

Let $Y$ be residual in $X$ . By [F2], fix one sequence $(N_{n})$ of nowhere dense subsets with $X ∖ Y \subseteq ⋃_{n} N_{n}$ . Since no nonempty open subset of a Baire space is meagre [F1], $Y$ is dense in $X$ .

2.1F1step 1.1

The sets $X ∖ C_{n}$ are dense open in $X$ . By the Baire property, their intersection meets every nonempty open subset $V$ of $O$ ; a point in that intersection and $V$ lies in every $G_{n}$ . Thus $O$ is Baire, including the empty case.

2.2F3step 1.2

Let $(G_{n})$ be dense open subsets of $Y$ . Put $F_{n} = Y ∖ G_{n}$ and $C_{n} = {\overline{F_{n}}}^{X}$ . Because $Y$ is dense, each $C_{n}$ is nowhere dense in $X$ : otherwise a nonempty $X$ -open $V \subseteq C_{n}$ would meet $Y$ , and the nonempty relatively open $V \cap Y$ would lie in $C_{n} \cap Y = F_{n}$ , contradicting density of $G_{n}$ in $Y$ .

3.1F1F2F3step 1.2step 2.2

The single interleaved sequence $N_{0}, C_{0}, N_{1}, C_{1}, \dots$ witnesses that $(X ∖ Y) \cup ⋃_{n} C_{n}$ is meagre; forming it requires no countable selection of witnesses. Given a nonempty relatively open $W = V \cap Y$ in $Y$ , $V$ is nonempty open in $X$ . By [F1], $V$ is not contained in that meagre set. Any point of $V$ outside it belongs to $Y$ and every $G_{n}$ , hence to $W \cap ⋂_{n} G_{n}$ . Therefore $Y$ is Baire.

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 establish both assertions.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)
Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)