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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The points polygonally reachable from a fixed point form a clopen subset of every open subset of $\mathbb{R}^n$

Statement

Let $U\subseteq\mathbb{R}^n$ be open and let $a\in U$ . The set $R_a$ of points of $U$ joined to $a$ by a polygonal path (Polygonal paths and polygonally connected subsets of $\mathbb{R}^n$ ) in $U$ is both open and closed in the subspace $U$ .

Facts & Assumptions

Given: An open subset $U\subseteq\mathbb{R}^n$ , a point $a\in U$ , and the polygonally reachable set $R_a\subseteq U$ .

[L1]

Every point of an open set in a metric topology has an open metric ball contained in that set (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).

[L2]

A straight segment between two points of an Euclidean ball stays in that ball, by the triangle inequality for the Euclidean norm ( $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it).

[L3]

A finite concatenation of such segments is a continuous polygonal path (A finite concatenation of straight segments in $\mathbb{R}^n$ is a continuous path).

Proof

technique · direct

1.1

Let $y\in R_a$ . Choose $r>0$ with $B(y,r)\subseteq U$ . For every $z\in B(y,r)$ , the segment from $y$ to $z$ lies in $B(y,r)$ by [L2].

L1L2choose

1.2

Let $y\in U\setminus R_a$ and choose $r>0$ with $B(y,r)\subseteq U$ . If some $z\in B(y,r)$ lay in $R_a$ , a path from $a$ to $z$ followed by the segment from $z$ to $y$ would put $y$ in $R_a$ , a contradiction.

L1L2L3choose

2.1

Concatenating a polygonal path from $a$ to $y$ with that segment gives a polygonal path from $a$ to $z$ in $U$ . Thus $B(y,r)\subseteq R_a$ , so $R_a$ is open in $U$ .

L3step 1.1

3.1

Hence $B(y,r)\subseteq U\setminus R_a$ , so the complement is open in $U$ . Therefore $R_a$ is clopen in $U$ .

step 2.1step 1.2∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 25 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

Connected space (standard reference, not scraped)
Locally connected space (Wikipedia) (standard reference, not scraped)

The points polygonally reachable from a fixed point form a clopen subset of every open subset of Rn\mathbb{R}^nRn