CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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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.
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Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology

Statement

Both of the following properties of topological spaces are hereditary (Hereditary, open-hereditary and closed-hereditary properties of topological spaces).

Metrizability (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). If $\mathcal{T}$ is induced by a metric $d$ on $X$ and $S \subseteq X$ , then the subspace topology $\mathcal{T}_S$ (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) is induced by the subspace metric $d_S = d \restriction (S \times S)$ (Isometry, isometric embedding, and the subspace metric on a subset). So a subspace of a metrizable space is metrizable, and a metric inducing its topology is available explicitly and not merely asserted to exist.
First countability (First countable space: a countable neighbourhood base at every point). If every point of $X$ has an at most countable neighbourhood base and $S \subseteq X$ , then every point of $S$ has an at most countable neighbourhood base in $(S, \mathcal{T}_S)$ , namely the family of traces on $S$ of the members of a base at that point in $X$ .

Claim 1 is a corollary in the strict sense: the identification of the subspace topology with the metric topology of the subspace metric is discharged inside Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, and nothing is reproved here.

Facts & Assumptions

Given: A topological space $(X, \mathcal{T})$ , a subset $S \subseteq X$ with the subspace topology $\mathcal{T}_S = \{\, U \cap S : U \in \mathcal{T} \,\}$ , and a point $x \in S$ .

[A1]

A space is metrizable when some metric on it has the given topology as its metric topology (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[A2]

$N$ is a neighbourhood of $x$ in a space when some open set $U$ of that space satisfies $x \in U \subseteq N$ ; a family $\mathcal{B}_x$ of neighbourhoods of $x$ is a neighbourhood base at $x$ when every neighbourhood of $x$ contains a member of it (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

[A3]

A space is first countable when every one of its points has an at most countable neighbourhood base (First countable space: a countable neighbourhood base at every point, Finite, countably infinite, countable, uncountable).

[L1]

For $A \subseteq X$ and a metric $d$ on $X$ , the subspace topology $\{\, U \cap A : U \in \mathcal{T}_d \,\}$ is exactly the metric topology of the subspace metric $d_A$ (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, subspaces bullet; Isometry, isometric embedding, and the subspace metric on a subset).

[L2]

$\mathcal{T}_S$ is a topology on $S$ , and its members are exactly the traces $U \cap S$ with $U \in \mathcal{T}$ (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, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[L3]

A nonempty set is at most countable if and only if it admits a surjection from $\mathbb{N}$ (A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$ , Finite, countably infinite, countable, uncountable).

[L4]

A property $P$ is hereditary when every subspace of every space satisfying $P$ satisfies $P$ (Hereditary, open-hereditary and closed-hereditary properties of topological spaces).

Proof

technique · direct

1.1

Let $X$ be metrizable and let $d$ be a metric on $X$ with $\mathcal{T}_d = \mathcal{T}$ ; such a $d$ exists by [A1].

A1choose

1.2

Let $X$ be first countable and let $\mathcal{B}_x$ be an at most countable neighbourhood base at $x$ in $X$ ; such a family exists by [A3].

A3choose

2.1

$\mathcal{B}_x$ is nonempty, since $X$ itself is a neighbourhood of $x$ and so contains a member of $\mathcal{B}_x$ .

A2step 1.2

2.2

By [L1] applied with $A := S$ , the family $\{\, U \cap S : U \in \mathcal{T}_d \,\}$ is the metric topology of $d_S$ ; and $\mathcal{T}_d = \mathcal{T}$ by step 1.1, so that family is $\mathcal{T}_S$ by [L2].

step 1.1L1L2

2.3

Put $\mathcal{B}^S_x := \{\, N \cap S : N \in \mathcal{B}_x \,\}$ . Each of its members is a neighbourhood of $x$ in $S$ : given $N \in \mathcal{B}_x$ there is $U \in \mathcal{T}$ with $x \in U \subseteq N$ , and then $U \cap S \in \mathcal{T}_S$ with $x \in U \cap S \subseteq N \cap S$ .

step 1.2A2L2

2.4

Every neighbourhood $M$ of $x$ in $S$ contains a member of $\mathcal{B}^S_x$ : fix $W \in \mathcal{T}_S$ with $x \in W \subseteq M$ and write $W = U \cap S$ with $U \in \mathcal{T}$ by [L2]; then $U$ is a neighbourhood of $x$ in $X$ , so some $N \in \mathcal{B}_x$ satisfies $N \subseteq U$ , and $N \cap S \subseteq U \cap S = W \subseteq M$ .

step 1.2A2L2

3.1

$\mathcal{B}^S_x$ is nonempty and at most countable: by step 2.1 and [L3] there is a surjection $s : \mathbb{N} \to \mathcal{B}_x$ , and $k \mapsto s(k) \cap S$ is then a surjection $\mathbb{N} \to \mathcal{B}^S_x$ , so [L3] applies again.

step 2.1L3

3.2

By step 2.2 the topology $\mathcal{T}_S$ is the metric topology of the metric $d_S$ on $S$ , so $(S,\mathcal{T}_S)$ is metrizable by [A1]; as $X$ and $S$ were arbitrary, metrizability is hereditary by [L4]. This is claim 1.

step 2.2A1L4

4.1

By steps 2.3, 2.4 and 3.1 the family $\mathcal{B}^S_x$ is an at most countable neighbourhood base at $x$ in $(S,\mathcal{T}_S)$ , and $x \in S$ was arbitrary, so $(S,\mathcal{T}_S)$ is first countable by [A3]; as $X$ and $S$ were arbitrary, first countability is hereditary by [L4]. This is claim 2.

step 2.3step 2.4step 3.1A2A3L4∎

Remarks

No choice principle is spent. The metric $d_S$ is a restriction, and the neighbourhood base $\mathcal{B}^S_x$ is the image of a given family under an explicit map, so the enumeration of step 3.1 is produced from a given enumeration rather than selected. The only selections in the proof are the single metric of step 1.1 and the single family of step 1.2, each of which exists by hypothesis for the one space under consideration.
Neither converse holds, and neither is claimed. A subspace of a non-metrizable space may perfectly well be metrizable, every one-point subspace being so; heredity is a statement in one direction only.
The metric is not canonical, and the topology is. Claim 1 produces a metric on $S$ , the restriction of the one chosen on $X$ ; a different metric on $X$ inducing the same topology restricts to a different metric on $S$ inducing the same subspace topology (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). What is hereditary is the existence of a metric, which is a property of the topology alone.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 89 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

Metrizable space (Wikipedia) (standard reference, not scraped)
First-countable space (Wikipedia) (standard reference, not scraped)
Hereditary property (Wikipedia) (standard reference, not scraped)