TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)verified 2026-07-29 (claude-fable-5)

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A connected, locally path-connected space is path-connected, because its path components are open

Statement

Let $X$ be a locally path-connected topological space (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point). Then:

Path components are open, hence clopen, hence unions of them are clopen.
Components and path components agree: $P (x) = C (x)$ for every $x \in X$ (Paths, path-connected spaces and path components, Connected components, quasicomponents, and totally disconnected spaces).
If $X$ is moreover connected (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets) then $X$ is path-connected.

Claim 3 is the statement in the title; claims 1 and 2 are what carry it, and both are worth having on their own. Local path-connectedness alone does not make a space path-connected — a two-point discrete space is locally path-connected and is not path-connected — so the connectedness hypothesis in claim 3 is not removable.

Facts & Assumptions

Given: A locally path-connected space $X$ , with subsets carrying the subspace topology (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).

[A1]

For every $x \in X$ and every open $U ∋ x$ there is an open path-connected $V$ with $x \in V \subseteq U$ ; in particular, taking $U = X$ , an open path-connected $V ∋ x$ exists (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[A2]

The path components partition $X$ ; $y \in P (x)$ exactly when a path in $X$ joins $x$ to $y$ ; a path-connected subset of $X$ containing $x$ is contained in $P (x)$ , since each of its points is joined to $x$ inside it and hence in $X$ (Paths, path-connected spaces and path components, 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).

[A3]

A union of open sets is open, and a set is open when each of its points has an open set around it inside it (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[A4]

A space is connected exactly when its only clopen subsets are $\emptyset$ and the whole space; a subset $A$ is connected exactly when the only subsets of $A$ clopen in $A$ are $\emptyset$ and $A$ ; the traces of open and of closed sets are the open and the closed sets of a subspace (For a topological space the following agree: no separation exists, the only clopen subsets are $\emptyset$ and $X$ , and every continuous map to the two-point discrete space is constant, claims 1 and 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).

[A5]

$C (x)$ is the largest connected subset of $X$ containing $x$ ; a path-connected space, and a path-connected subset, is connected (Connected components, quasicomponents, and totally disconnected spaces, Every path-connected space is connected, and every path component lies inside a component, claim 2).

Proof

technique · direct

1.1

Let $x \in X$ and let $y \in P (x)$ . By [A1] there is an open path-connected $V$ with $y \in V \subseteq X$ , and $V \subseteq P (y) = P (x)$ by [A2], the set $V$ being path-connected and containing $y$ , which lies in $P (x)$ .

A1A2

1.2

For claim 2, $P (x) \subseteq C (x)$ by [A5], $P (x)$ being a path-connected subset containing $x$ , hence connected, and $C (x)$ the largest such.

A2A5

2.1

So every point of $P (x)$ has an open set around it inside $P (x)$ , whence $P (x)$ is open in $X$ by [A3].

step 1.1A3

3.1

$P (x)$ is also closed: its complement is the union of the remaining path components, which partition $X$ by [A2], and each of them is open by step 2.1; so the complement is open by [A3]. Hence every path component is clopen, and so is any union of them, being a union of open sets with complement a union of open sets. This is claim 1.

step 2.1A2A3

4.1

Conversely $P (x) \cap C (x)$ is clopen in the subspace $C (x)$ by step 3.1 and [A4], being the trace on $C (x)$ of a clopen subset of $X$ , and it is nonempty, containing $x$ ; since $C (x)$ is connected, [A4] forces $P (x) \cap C (x) = C (x)$ , that is $C (x) \subseteq P (x)$ .

step 3.1A4A5

5.1

Claim 2 follows from steps 1.2 and 4.1.

step 1.2step 4.1

6.1

For claim 3 assume $X$ is connected. If $X = \emptyset$ it is path-connected by [A2], having no pair of points to join. Otherwise fix $x \in X$ ; then $P (x)$ is clopen by step 3.1 and nonempty, so $P (x) = X$ by [A4], which says exactly that every point of $X$ is joined to $x$ by a path, and hence any two points are joined to each other. So $X$ is path-connected.

step 3.1A2A4∎

Remarks

Why the argument is about path components and not about paths. The hypothesis gives small open path-connected sets, and the only use made of them is that they cannot straddle two path components. That turns a local statement into the global partition of claim 1 with no construction of a long path anywhere; the path joining two given points is produced only at the very end, by the definition of $P (x)$ .
Claim 2 is why local path-connectedness is the right hypothesis in practice. Under it the two partitions of $X$ coincide, so "connected" and "path-connected" become interchangeable for subspaces that are open, and every connectedness computation can be done with paths.
What fails without local path-connectedness. Claim 1 is exactly where the hypothesis is spent: without it a path component need not be open, its complement need not be open, and the clopen argument collapses. A connected space whose path components are not open, and which is therefore connected and not path-connected, is constructed later on this page.

Depends on

Used by

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Sources

Locally connected space (Wikipedia) (standard reference, not scraped)
Paul Bankston, Metric Topology: A First Course (standard reference, not scraped)