TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1givenF2F1

The empty subspace has its unique compatible complete metric.

2.1step 1.1F3F2

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

3.1step 2.1F3F2

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.

4.1step 3.1∎

The preceding construction and implications establish the assertion.

Depends on

Used by

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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