TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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For an open subset of $\mathbb{R}^n$ , connectedness, path-connectedness and polygonal connectedness are equivalent

Statement

Let $U\subseteq\mathbb{R}^n$ be open. Then $U$ is connected if and only if it is path-connected, if and only if it is polygonally connected.

Facts & Assumptions

Given: An open subset $U\subseteq\mathbb{R}^n$ .

[L1]

The polygonally reachable set from a point of $U$ is clopen in $U$ (The points polygonally reachable from a fixed point form a clopen subset of every open subset of $\mathbb{R}^n$ ).

[L2]

A polygonal path is a path, and every path-connected space is connected (Polygonal paths and polygonally connected subsets of $\mathbb{R}^n$ , Every path-connected space is connected, and every path component lies inside a component).

[L3]

A connected space has no nonempty proper clopen subset (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Proof

technique · direct

1.1

Suppose $U$ is connected. If $U=\varnothing$ , then polygonal connectedness, path-connectedness, and connectedness all hold vacuously. Otherwise choose $a\in U$ . The reachable set $R_a$ is nonempty and clopen by [L1], so [L3] gives $R_a=U$ .

L1L3caseschoose

1.2

Polygonal connectedness implies path-connectedness, and path-connectedness implies connectedness, by [L2].

2.1

In the nonempty case, every point of $U=R_a$ is joined to $a$ by a polygonal path; reversing one such path and concatenating it with another joins any two points of $U$ . Together with the empty case, connectedness implies polygonal connectedness.

step 1.1

3.1

Steps 2.1 and 1.2 give all three equivalences.

step 2.1step 1.2∎

Depends on

Used by

Every connected component of an open subset of ℝⁿ is open and polygonally connected Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 78 results over 15 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

Path-connected space (standard reference, not scraped)
Connected space (Wikipedia) (standard reference, not scraped)

For an open subset of Rn\mathbb{R}^nRn, connectedness, path-connectedness and polygonal connectedness are equivalent