False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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FALSE: every retract is a deformation retract

Statement

False claim. Every retract of a topological space is a deformation retract.

Facts & Assumptions

Given: The two-point set $X=\{0,1\}$ with the discrete topology and its singleton subspace $A=\{0\}$ .

[A1]

A retraction $r:X\to A$ satisfies $r(a)=a$ on $A$ ; a deformation retraction additionally supplies a homotopy from $\operatorname{id}_X$ to the inclusion followed by $r$ , fixed on $A$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[L1]

Every map out of a discrete space is continuous, since every subset of its domain is open (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[L2]

A nonempty space whose identity is nullhomotopic is contractible, and every nonempty contractible space is path-connected (A nonempty space is contractible if and only if its identity map is nullhomotopic, Every nonempty contractible space is path-connected, Paths, path-connected spaces and path components).

[L3]

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

[L4]

A separation is a pair of disjoint nonempty open sets whose union is the space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Refutation

technique · direct

1.1

Define $r:X\to A$ by $r(0)=r(1)=0$ . This map is continuous by [L1] and satisfies $r(0)=0$ , so it is a retraction by [A1].

L1A1

1.2

Suppose $A$ were a deformation retract of $X$ . Since the inclusion followed by $r$ is the constant map $c_0:X\to X$ , [A1] would give $\operatorname{id}_X\simeq c_0$ . Then $X$ would be contractible, hence path-connected by [L2], and hence connected by [L3].

assume-hypA1L2L3

1.3

But $\{0\}$ and $\{1\}$ are disjoint nonempty open subsets of the discrete space $X$ and their union is $X$ , so they form a separation by [L4]. Thus $X$ is disconnected.

L1L4

2.1

Steps 1.2 and 1.3 contradict one another. Hence $A$ is a retract of $X$ by step 1.1 but is not a deformation retract, refuting the claim.

step 1.1step 1.2step 1.3∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 103 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)