Algebraic Topology · Problem 6

The Fundamental Group — Allen Hatcher

Mathematics — Topology · deformation-retraction

Problem

(a) Let

X

be the subspace of

\mathbb{R}^2

consisting of the horizontal segment

[0,1] \times \{0\}

together with all the vertical segments

\{r\} \times [0, 1-r]

for

r

a rational number in

[0,1]

. Show that

X

deformation retracts to any point in the segment

[0,1] \times \{0\}

, but not to any other point. [See the preceding problem.] (b) Let

Y

be the subspace of

\mathbb{R}^2

that is the union of an infinite number of copies of

X

arranged as in the figure below. Show that

Y

is contractible but does not deformation retract onto any point. (c) Let

Z

be the zigzag subspace of

Y

homeomorphic to

\mathbb{R}

indicated by the heavier line. Show there is a deformation retraction in the weak sense (see Exercise 4) of

Y

onto

Z

, but no true deformation retraction.