TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Under the Axiom of Countable Choice, every $G_{δ}$ subspace of a complete metric space is completely metrizable

Statement

Assume the Axiom of Countable Choice. If $(X, d)$ is a complete metric space and $Y \subseteq X$ is $G_{δ}$ in $X$ , then the subspace $Y$ is completely metrizable.

Facts & Assumptions

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

[F1]

Let $(X, T)$ be a topological space (def-topological-space) and let $A \subseteq X$ . $A$ is a $G_{δ}$ set of $X$ when there is a sequence $(V_{n})_{n \in N}$ of open subsets of $X$ with $A = ⋂_{n \in N} V_{n}$ , and an $F_{σ}$ set of $X$ when there is a sequence $(F_{n})_{n \in N}$ of closed subsets of $X$ with $A = ⋃_{n \in N} F_{n}$ . ( $G_{δ}$ and $F_{σ}$ subsets of a topological space, agreeing with the real-line notion).

[F2]

If $X$ is completely metrizable and $U \subseteq X$ is open, then $U$ is completely metrizable in its subspace topology. (Every open subspace of a completely metrizable space is completely metrizable).

[F3]

Assume the Axiom of Countable Choice. If $(X, d)$ is metrizable and $(Y_{n})_{n \in N}$ is a sequence of completely metrizable subspaces of $X$ , then $⋂_{n} Y_{n}$ is completely metrizable. (Under the Axiom of Countable Choice, a countable intersection of completely metrizable subspaces is completely metrizable).

Proof

technique · direct

1.1

The empty subspace has its unique compatible complete metric.

givenF2F1

2.1

Otherwise write the subspace as a countable intersection of open subspaces.

step 1.1F3F2

3.1

Each open subspace is completely metrizable by the reciprocal-distance lemma, and the countable-intersection metric then gives a compatible complete metric on the intersection.

step 2.1F3F2

4.1

The preceding construction and implications establish the assertion.

step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 88 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

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)

Under the Axiom of Countable Choice, every GδGδ​ subspace of a complete metric space is completely metrizable