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Problem 71B
Problem 71B
#
Problem 71B
(Brouwer fixed point theorem)
.
Use the previous problem to prove that any continuous
function
𝑓
:
𝐷
𝑛
→
𝐷
𝑛
has a fixed point.
Solution
by
kiwiyou
Assume that
𝑓
has no fixed point. We can define
𝐹
(
𝑥
)
as the intersection of
𝑆
𝑛
−
1
and the ray from
𝑓
(
𝑥
)
to
𝑥
. The intersection is uniquely determined since
𝑥
≠
𝑓
(
𝑥
)
.
𝐷
𝑛
𝑆
𝑛
−
1
𝑥
𝑓
(
𝑥
)
𝐹
(
𝑥
)
If such
𝑓
exists, the composition
𝑆
𝑛
−
1
↪
𝐷
𝑛
→
𝐹
𝑆
𝑛
−
1
is identity map, which is a contradiction.
∎