CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-07-31

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.

Every nonempty contractible space is path-connected

Statement

Every nonempty contractible topological space is path-connected.

Facts & Assumptions

Given: A nonempty contractible space $X$ and points $x, y \in X$ .

[L1]

The identity of $X$ is homotopic to a constant map $c_{x_{0}}$ for some $x_{0} \in X$ (A nonempty space is contractible if and only if its identity map is nullhomotopic, Nullhomotopic maps and contractible spaces).

[A1]

Paths define an equivalence relation: paths may be reversed and concatenated, and $X$ is path-connected exactly when every pair of points is joined by a path (Paths, path-connected spaces and path components).

[L2]

Product projections are continuous, and a map into a product is continuous exactly when its components are continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice).

[L3]

A map is continuous exactly when preimages of open sets are open (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ ).

Proof

technique · direct

1.1

Let $H : X \times I \to X$ be a homotopy from ${id}_{X}$ to $c_{x_{0}}$ . For each $z \in X$ , the map $j_{z} : I \to X \times I$ , $j_{z} (t) = (z, t)$ , is continuous by [L2], its components being constant and the identity.

L1L2

2.1

The map $γ_{z} : = H \circ j_{z} : I \to X$ is continuous because $(H \circ j_{z})^{- 1} [V] = j_{z}^{- 1} [H^{- 1} [V]]$ is open for every open $V \subseteq X$ . It has $γ_{z} (0) = z$ and $γ_{z} (1) = x_{0}$ , so it is a path from $z$ to $x_{0}$ .

step 1.1L1L3A1

3.1

Step 2.1 gives a path from $x$ to $x_{0}$ and a path from $y$ to $x_{0}$ . Reversing the latter and concatenating it with the former gives a path from $x$ to $y$ by [A1].

step 2.1A1

4.1

Since $x, y \in X$ were arbitrary, $X$ is path-connected.

step 3.1A1∎

Depends on

Used by

FALSE: every retract is a deformation retract False statement
The quaternion double cover generates the third homotopy group of SO(3) Lemma
Every nonempty convex subset of ℝⁿ is simply connected Theorem

Dependency tree · two levels

28 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

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)
Homotopy lecture notes (University of Padua) (standard reference, not scraped)