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

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For every space $X$ , the cylinder $X\times[0,1]$ deformation retracts onto $X\times\{0\}$

Example

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

$r(x,t)=(x,0),\qquad K((x,t),s)=(x,(1-s)t)$

form a deformation retraction of $Z$ onto $A$ .

Facts & Assumptions

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

[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 $\prod_{i \in I} X_i$ 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 $I\subseteq\mathbb R$ are continuous (For continuous maps into a convex subset of $\mathbb{R}^n$ , 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 $f(\overline{A}) \subseteq \overline{f(A)}$ ).

[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 $r:Z\to A$ , $r(x,t)=(x,0)$ , is continuous: its composite with the inclusion $A\hookrightarrow Z$ has the continuous components $(x,t)\mapsto x$ and the constant $0$ , so [L1] applies. It fixes every $(x,0)\in A$ , hence is a retraction.

L1A1

1.2

On the source $Z$ , the second projection $p_I:Z\to I$ and the constant zero map are continuous. Since $I$ is convex, [L2] makes $L:Z\times I\to I$ , $L((x,t),s)=(1-s)t$ , 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 $s\in I$ .

algebra

2.1

The first component $((x,t),s)\mapsto x$ 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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 74 results over 15 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)

For every space XXX, the cylinder X×[0,1]X\times[0,1]X×[0,1] deformation retracts onto X×{0}X\times\{0\}X×{0}