TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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

Statement

Let $U \subseteq R^{n}$ 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 $U \subseteq R^{n}$ .

[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 $R^{n}$ ).

[L2]

A polygonal path is a path, and every path-connected space is connected (Polygonal paths and polygonally connected subsets of $R^{n}$ , 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 $U = \emptyset$ , 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

Dependency tree · two levels

23 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

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

For an open subset of RnRn, connectedness, path-connectedness and polygonal connectedness are equivalent