CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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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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A discrete embedded submanifold is locally closed and countable

Statement

Let $S$ be a discrete embedded submanifold of a smooth manifold $M$ . Then every point of $S$ has a neighbourhood $U$ in $M$ such that $S \cap U$ is closed in $U$ . Moreover $S$ is countable.

Facts & Assumptions

Given: A discrete embedded submanifold $S \subseteq M$ .

[F1]

An embedded submanifold is locally a coordinate slice (Embedded submanifolds and slice charts).

[L1]

A subspace of a second-countable space is second countable (Second countability is hereditary, Second countability: an at most countable basis for the topology, 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

1.1

Because $S$ is discrete, its local dimension is $0$ . Thus by [F1], around each point $p \in S$ there is a chart $U$ in which $S \cap U$ corresponds to $R^{0} \times {0}$ , that is, a single point. A singleton is closed in the chart domain, so $S \cap U$ is closed in $U$ .

F1given

2.1

The manifold $M$ is second countable, hence so is its subspace $S$ by [L1]. Let $B$ be a countable basis for $S$ . Because $S$ is discrete, every singleton ${x}$ is open. Applying the basis property to the open set ${x}$ shows that some $B \in B$ satisfies $x \in B \subseteq {x}$ , hence $B = {x}$ . Thus every singleton of $S$ is itself a member of the countable family $B$ , so $S$ is countable.

L1step 1.1

3.1

Therefore $S$ is locally closed and countable.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

John M. Lee, Introduction to Smooth Manifolds, Embedded Submanifolds (standard reference, not scraped)