Alphabeta MathHow 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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise
Definition
Let carry the subspace topology, and let be the inclusion (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).
A continuous map is a retraction of onto if
equivalently, if for every . When such an exists, is a retract of .
The subspace is a deformation retract of if there are a retraction and a homotopy
Thus , , and for all and (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints). The pair is a deformation retraction.
Some sources call the pointwise-fixed condition a strong deformation retract. In this library the word deformation retract always includes it.
Depends on
- Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
- Continuity of a map of topological spaces at a point and globally
- 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
Used by
- Every nonempty retract of a contractible space is contractible Corollary
- A singleton is a retract but not a deformation retract of the two-point discrete space Example
- For every space X, the cylinder X×[0,1] deformation retracts onto X×{0} Example
- The Hawaiian earring retracts onto each of its circles Example
- π₁(ℝ²∖{0})≅ℤ Example
- FALSE: every retract is a deformation retract False statement
- A fixed-point-free self-map of the disk produces a continuous retraction onto the unit circle Lemma
- Finite wedges of quotient circles have van Kampen covers at the wedge point Lemma
- A retract induces an injection on fundamental groups, and a deformation retract induces an isomorphism Proposition
- For n≥1, radial normalisation is a deformation retraction of ℝⁿ∖{0} onto Sⁿ⁻¹ Theorem
- Mapping cylinder factorization Theorem
- The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse Theorem
Dependency tree · two levels
11 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)