CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-05 (claude-sonnet-5)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every connected component of an open subset of $\mathbb{R}^n$ is open and polygonally connected

Statement

Every connected component of an open subset $U\subseteq\mathbb{R}^n$ is open in $\mathbb{R}^n$ and polygonally connected.

Facts & Assumptions

Given: An open subset $U\subseteq\mathbb{R}^n$ and a connected component $C$ of $U$ .

[L1]

A Euclidean ball is path-connected, hence connected: the norm triangle inequality keeps every straight segment in the ball, the segment is continuous, and every path-connected space is connected (Open ball, closed ball and sphere in a metric space, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, A finite concatenation of straight segments in $\mathbb{R}^n$ is a continuous path, Every path-connected space is connected, and every path component lies inside a component).

[L2]

A component is the largest connected subset containing each of its points (Connected components, quasicomponents, and totally disconnected spaces).

[L3]

An open connected Euclidean subset is polygonally connected (For an open subset of $\mathbb{R}^n$ , connectedness, path-connectedness and polygonal connectedness are equivalent).

Proof

technique · direct

1.1

Let $x\in C$ . Since $U$ is open, choose $r>0$ with $B(x,r)\subseteq U$ . The ball is connected by [L1], meets $C$ at $x$ , and so lies in $C$ by maximality of the component.

L1L2choose

2.1

Therefore every point of $C$ has a Euclidean ball contained in $C$ , so $C$ is open in $\mathbb{R}^n$ .

step 1.1

3.1

The component $C$ is connected and now open, so [L3] makes it polygonally connected.

L3step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 103 results over 22 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

Locally connected space (standard reference, not scraped)

Every connected component of an open subset of Rn\mathbb{R}^nRn is open and polygonally connected