False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

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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.

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FALSE: every connected space is locally connected

Statement

False claim: every connected topological space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets) is locally connected (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).

Neither condition implies the other, and this item refutes one of the two directions. The other fails as well: a two-point discrete space is locally connected and disconnected, as Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point records.

Witness. The closure $\overline{G}$ of the zigzag graph (The graph of the piecewise-linear map oscillating between $0$ and $1$ on the intervals $[1/(n+2), 1/(n+1)]$ is path-connected, its closure adds the segment $\{0\} \times [0,1]$ , and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected), a subspace of $\mathbb{R}^2$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), is connected and is not locally connected at any point of the segment $\{0\} \times [0,1]$ .

Facts & Assumptions

Given: The zigzag graph $G \subseteq \mathbb{R}^2$ and its closure $\overline{G}$ , with the subspace topology.

[A1]

$X$ is locally connected when for every $x \in X$ and every open $U \ni x$ there is an open connected $V$ with $x \in V \subseteq U$ ; being locally connected requires this at every point (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[L1]

$\overline{G}$ is connected (The graph of the piecewise-linear map oscillating between $0$ and $1$ on the intervals $[1/(n+2), 1/(n+1)]$ is path-connected, its closure adds the segment $\{0\} \times [0,1]$ , and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected, claim 3).

[L2]

$\overline{G}$ is not locally connected at any point $(0,t)$ with $t \in [0,1]$ , and such points belong to $\overline{G}$ (The graph of the piecewise-linear map oscillating between $0$ and $1$ on the intervals $[1/(n+2), 1/(n+1)]$ is path-connected, its closure adds the segment $\{0\} \times [0,1]$ , and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected, claims 2 and 5).

Refutation

technique · contradiction

1.1

Suppose, for contradiction, that the claim holds: every connected space is locally connected.

assume-contra

2.1

$\overline{G}$ is connected by [L1], so the supposed claim applies to it and $\overline{G}$ is locally connected.

step 1.1L1

3.1

By [A1] this means $\overline{G}$ is locally connected at every one of its points, in particular at $(0,0)$ , which lies in $\overline{G}$ by [L2].

step 2.1A1L2

4.1

This contradicts [L2], which denies local connectedness at that point. So the claim is false.

step 3.1L2discharge-contradiction∎

Remarks

Connectedness is global and local connectedness is not, so no implication is to be expected in either direction. Connectedness says the space cannot be cut in two; local connectedness says every point has arbitrarily small connected open neighbourhoods. A space can be a single unbroken piece and still be locally shredded at some of its points, which is what $\overline{G}$ is at every point of the added segment.
What the failure costs. By A space is locally connected exactly when every component of every open subspace is open; in that case the components of the space itself are clopen and Connected components, quasicomponents, and totally disconnected spaces, a locally connected space has clopen components and, inside every open set, open components. In $\overline{G}$ that machinery is unavailable, which is precisely why its partition into path components fails to be a partition into clopen pieces and why the space is connected without being path-connected.
The failure is confined to the segment. Claim 5 of The graph of the piecewise-linear map oscillating between $0$ and $1$ on the intervals $[1/(n+2), 1/(n+1)]$ is path-connected, its closure adds the segment $\{0\} \times [0,1]$ , and that closure is connected, is not path-connected because no path joins the segment to the graph, and is not locally connected locates it at the points $(0,t)$ , and at every other point $\overline{G}$ agrees locally with $G$ , which is locally connected by claim 1 there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 104 results over 17 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 (Wikipedia) (standard reference, not scraped)
Connected space (Wikipedia) (standard reference, not scraped)
Paul Bankston, Metric Topology: A First Course (standard reference, not scraped)
Topologist's sine curve (Wikipedia) (standard reference, not scraped)