Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 21F / b

Solution

A subspace

Y \subseteq X

X

if there is a continuous retraction

r : X \to Y

such that

r (y) = y

for every

y \in Y

. A space is a contractible space if its identity map is homotopic to a constant map.

Let

i : Y ↪ X

be the inclusion and let

H : X \times I \to X

be a contraction, with

H (x, 0) = x

and

H (x, 1) = x_{0}

. Then

K (y, t) = r (H (i (y), t))

is continuous and satisfies

K (y, 0) = r (i (y)) = y, K (y, 1) = r (x_{0}) .

Thus $K$ contracts $Y$ , proving that every retract of a contractible space is contractible.

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