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ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31
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For every space X, the cylinder X×[0,1] deformation retracts onto X×{0}

Example

For every topological space X, put Z=X×I and A=X×{0}. The maps

r(x,t)=(x,0),K((x,t),s)=(x,(1−s)t)

form a deformation retraction of Z onto A.

Facts & Assumptions

Given: A topological space X, the product Z=X×I, and its subspace A=X×{0}.

[L1]

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, The product set ∏i∈IXi of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

[L2]

Straight-line homotopies between continuous maps into the convex interval I⊆R are continuous (For continuous maps into a convex subset of Rn, the straight-line formula defines a continuous homotopy).

[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(A‾)⊆f(A)‾).

[A1]

A deformation retraction consists of a retraction and a homotopy from the identity to the inclusion followed by it, fixed pointwise on the retract (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

Verification

technique · direct
1.1

The map r:Z→A, r(x,t)=(x,0), is continuous: its composite with the inclusion A↪Z has the continuous components (x,t)↦x and the constant 0, so [L1] applies. It fixes every (x,0)∈A, hence is a retraction.

L1A1
1.2

On the source Z, the second projection pI:Z→I and the constant zero map are continuous. Since I is convex, [L2] makes L:Z×I→I, L((x,t),s)=(1−s)t, continuous.

L1L2
1.3

One has K((x,t),0)=(x,t), K((x,t),1)=(x,0), and K((x,0),s)=(x,0) for every s∈I.

algebra
2.1

The first component ((x,t),s)↦x is continuous as a composite of product projections, since the preimage of an open set is an iterated preimage and hence open by [L3]. Together with step 1.2, [L1] makes K((x,t),s)=(x,L((x,t),s)) continuous into Z.

step 1.2L1L3
3.1

Steps 1.1, 2.1 and 1.3 satisfy [A1], so (r,K) is a deformation retraction of Z onto A.

step 1.1step 2.1step 1.3A1∎

Depends on

Used by

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Dependency tree · two levels

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Sources