False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

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.

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 $R^{2}$ (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 $G \subseteq R^{2}$ and its closure $\overline{G}$ , with the subspace topology.

[A1]

$X$ is locally connected when for every $x \in X$ and every open $U ∋ x$ 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 · two levels

39 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

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)