TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)verified 2026-07-29 (claude-fable-5)

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Every path-connected space is connected, and every path component lies inside a component

Statement

Let $X$ be a topological space, 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). Then:

The unit interval is connected. $I = [0, 1]$ is a connected subset of $R$ , hence a connected space.
Path-connected implies connected. If $X$ is path-connected (Paths, path-connected spaces and path components) then $X$ is connected (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets). The same holds for a subset: a path-connected subset of $X$ is a connected subset of $X$ .
Path components refine components. For every $x \in X$ , $P (x) \subseteq C (x),$ the path component inside the component (Connected components, quasicomponents, and totally disconnected spaces). So every component is a union of path components.

No converse is claimed. Claim 2 is one-directional and claim 3 is an inclusion; the question of when a connected space is path-connected is not settled here.

No choice principle is used. The proof takes the union over the set of all paths issuing from a fixed point rather than selecting one path per endpoint, which is what an appeal to the Axiom of Choice would be. The point at which the temptation arises is flagged in the remarks.

Facts & Assumptions

Given: A topological space $X$ and the unit interval $I = [0, 1]$ with the subspace topology from $R$ (Paths, path-connected spaces and path components).

[A1]

A subset of $R$ is a connected subset exactly when it is order-convex, and $[0, 1]$ is order-convex (The connected subspaces of $R$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $R$ ", Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[A2]

A continuous image of a connected space is a connected subset of the target (A continuous image of a connected space is connected, and connectedness is a topological property, claim 1).

[A3]

A union of connected subsets with a point in common is connected (A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member, claim 1).

[A4]

A path in $X$ from $x$ to $y$ is a continuous map $γ : I \to X$ with $γ (0) = x$ and $γ (1) = y$ ; $X$ is path-connected when every pair of its points is joined by one; the path component $P (x)$ is the set of points joined to $x$ , and it is a path-connected subset of $X$ (Paths, path-connected spaces and path components, Continuity of a map of topological spaces at a point and globally).

[A5]

$C (x)$ is the largest connected subset of $X$ containing $x$ ; the empty space is connected (Connected components, quasicomponents, and totally disconnected spaces, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Proof

technique · direct

1.1A1

$[0, 1]$ is order-convex, so it is a connected subset of $R$ by [A1], that is the space $I$ is connected; this is claim 1.

1.2A5given

Assume $X$ is path-connected. If $X = \emptyset$ it is connected by [A5] and claim 2 holds, so assume $X \neq \emptyset$ and fix a point $x_{0} \in X$ .

1.3A4

Let $Γ : = {γ : γ is a path in X with γ (0) = x_{0}}$ , a set of functions from $I$ to $X$ . No member of $Γ$ is selected: the whole family is used.

2.1step 1.1step 1.3A2A4

For each $γ \in Γ$ the image $γ [I]$ is a connected subset of $X$ , by step 1.1 and [A2] applied to the continuous map $γ$ ; and $x_{0} = γ (0) \in γ [I]$ .

2.2step 1.2step 1.3A4

$X = ⋃_{γ \in Γ} γ [I]$ : each image is a subset of $X$ , and conversely every $y \in X$ is joined to $x_{0}$ by some path $γ$ , which lies in $Γ$ and has $y = γ (1) \in γ [I]$ .

3.1step 2.1step 2.2A3

Hence $X$ is connected by [A3], being a union of connected sets all containing $x_{0}$ . Applied to the space $A$ with its subspace topology, the same argument shows that a path-connected subset $A \subseteq X$ is a connected subset of $X$ ; this is claim 2.

4.1step 3.1A4A5∎

For claim 3, $P (x)$ is a path-connected subset of $X$ by [A4], hence a connected subset of $X$ by claim 2, and it contains $x$ ; so $P (x) \subseteq C (x)$ by the maximality in [A5]. Since the path components partition $X$ by [A4] and each lies inside a single component, every component is a union of path components.

Remarks

Where choice would have crept in. The textbook phrasing "for each $y \in X$ choose a path from $x_{0}$ to $y$ " produces a family of paths indexed by $X$ and is an application of the Axiom of Choice over an arbitrary index set. It is unnecessary: the union of the images of all paths from $x_{0}$ is already $X$ , and forming that union selects nothing. Step 1.3 is written to make the difference visible rather than to leave it to the reader.
Claim 1 is where the real line enters, and it enters once. Everything else in the proof is formal. All the content of "path-connected implies connected" is the connectedness of the interval, which is a consequence of the least upper bound property through The connected subspaces of $R$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $R$ ".
Claim 3 gives the standard picture. Components are unions of path components, so the two partitions of $X$ are nested, with the path components the finer of the two. They coincide in many familiar spaces and not in all, and nothing above says which case a given space is in.

Depends on

Used by

Diameter rigidity from toponogov under a sectional lower bound Corollary
Every connected component of an open subset of ℝⁿ is open and polygonally connected Corollary
For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected Corollary
Hartogs extension by a compact-support dbar correction Corollary
ℝⁿ is polygonally connected, connected, locally path-connected and locally connected Corollary
Singular cohomology satisfies the Eilenberg Steenrod cohomology axioms Corollary
A complete manifold with zero global injectivity radius Counterexample
A connected complex domain need not be star-shaped Counterexample
A connected plane domain that is not homologically simply connected Counterexample
A curl-free C¹ field on the complement of a line that is not conservative Counterexample
A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
An isolated boundary point obstructs pointwise-zero Green data Counterexample
Antipodal points on a round sphere have many minimizing geodesics Counterexample
The comb space is path-connected and fails to be locally connected at every point of the limit tooth strictly above the base, so path-connectedness does not imply local connectedness Counterexample
Quaternionic clutching bundles ξ_h,j over S⁴ Definition
The Jacobian of a compact Riemann surface Definition
A genus-two compact surface gives a cocompact Fuchsian group Example
A Hartogs domain with a strictly plurisubharmonic exhaustion Example
A strictly plurisubharmonic exhaustion of the convex unit ball Example
Atlases on the sphere, plane, disc and annulus Example
Cartan hadamard for hyperbolic space Example
Cut locus on a flat rectangular torus from the Dirichlet cell Example
Cutoff extension across a puncture in complex dimension two Example
Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected Example
Every cycle in a round annulus has one period, that of the central circle Example
First Cousin gluing on the pseudoconvex domain ℂ Example
Hopf–Rinow on a flat cylinder Example
Hyperbolic space is complete Example
Mod-two duality for real projective space Example
Rectangular lattices, real mapping, and inverse elliptic integrals Example
The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies placed in the connectedness hierarchy Example
The hexagonal A₂ cell: Euclidean cell metric versus graph distance Example
FALSE: every connected topological space is path-connected False statement
FALSE: every retract is a deformation retract False statement
FALSE: the intersection of two connected subspaces is connected False statement
Positive ricci curvature without a uniform lower bound implies compactness False statement
The distance from p is smooth on m minus p False statement
A disc missing p carries a holomorphic logarithm of z-p Lemma
A finite endpoint of a maximal unit-speed geodesic produces a Cauchy curve Lemma
A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma

…and 41 more results.

Dependency tree · two levels

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Sources

Connected space (Wikipedia) (standard reference, not scraped)
Keith Conrad, Spaces That Are Connected but Not Path-Connected (standard reference, not scraped)