ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

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A singleton is a retract but not a deformation retract of the two-point discrete space

Example

Let $X = {0, 1}$ have the discrete topology and let $A = {0}$ . The constant map $r : X \to A$ is a retraction, but $A$ is not a deformation retract of $X$ .

Facts & Assumptions

Given: The two-point discrete space $X$ and its singleton subspace $A$ .

[L1]

Every map out of a discrete space is continuous (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[A1]

A deformation retraction would give a homotopy from ${id}_{X}$ to the constant map at $0$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[L2]

The refutation in FALSE: every retract is a deformation retract proves that this $A$ is a retract of $X$ but not a deformation retract, because such a deformation would force the disconnected space $X$ to be path-connected (Paths, path-connected spaces and path components).

Verification

technique · direct

1.1

The map $r (0) = r (1) = 0$ is continuous by [L1] and fixes the point of $A$ , so it is a retraction.

L1A1

1.2

If a deformation retraction existed, [A1] would make ${id}_{X}$ homotopic to the constant map at $0$ ; the full contradiction with the separation $X = {0} \cup {1}$ is established in [L2].

assume-hypA1L2

2.1

Therefore $A$ is a retract but not a deformation retract of $X$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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)