PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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Topological manifolds are locally compact and locally path connected

Statement

Every topological manifold is locally compact and locally path connected. More precisely, every point has a neighbourhood basis consisting of open sets whose closures are compact and which are path connected.

Facts & Assumptions

Given: A topological manifold $M$ and a point $p \in M$ .

[F1]

Every neighbourhood of $p$ contains a coordinate ball $φ^{- 1} [B (c, r)]$ whose closure is compact (Coordinate balls form a basis of a topological manifold).

[L1]

For $n \geq 1$ , every Euclidean open ball is path connected and every neighbourhood in $R^{n}$ contains a path-connected open ball ( $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected).

[L2]

For $n \geq 1$ , Euclidean space is locally compact ( $R^{n}$ is locally compact and $σ$ -compact).

[A1]

Homeomorphisms preserve path connectedness and compactness.

Proof

technique · direct

1.1

Let $O$ be any open neighbourhood of $p$ . By [F1] choose a chart [F1, choose] $(U, φ)$ at $p$ and an open Euclidean ball $B (c, r)$ such that $p \in φ^{- 1} [B (c, r)] \subseteq O$ and the closure of $φ^{- 1} [B (c, r)]$ in $M$ is compact. This already gives a compact neighbourhood basis at $p$ , so $M$ is locally compact.

F1choose

2.1

If the manifold dimension is $n = 0$ , then every point is open, so $M$ is [L1, A1, step 1.1] locally path connected trivially. If $n \geq 1$ , then [L1] says the Euclidean ball $B (c, r)$ is path connected. Since $φ$ is a homeomorphism on $U$ , [A1] makes $φ^{- 1} [B (c, r)]$ path connected. Thus every neighbourhood of $p$ contains an open path-connected neighbourhood of $p$ .

L1A1step 1.1

3.1

Step 1.1 proves local compactness and step 2.1 proves local path [step 1.1, step 2.1, L2] connectedness. The role of [L2] is only to justify that the compact-neighbourhood conclusion in step 1.1 matches the Euclidean local model used to produce the coordinate balls.

step 1.1step 2.1L2∎

Depends on

Used by

Components of a topological manifold are open and at most countable Proposition
Topological manifolds are metrizable and paracompact Theorem

Dependency tree · two levels

16 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

Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.2 (standard reference, not scraped)