 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/21f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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.

 New to topics? Read the docs here!