218 CHAPTER 10. BROUWER FIXED POINT THEOREM Rn∗
Lemma 10.2.3 Let f : B(p,r)→ Rn where the ball is also in Rn. Let f be one to one, fcontinuous. Then there exists δ > 0 such that
f(
B(p,r))⊇ B(f(p) ,δ ) .
In other words, f(p) is an interior point of f(
B(p,r))
.
Proof: Since f(
B(p,r))
is compact, it follows that f−1 : f(
B(p,r))→ B(p,r) is con-
tinuous. By Lemma 10.2.2, there exists a polynomial g : f(
B(p,r))→ Rn such that∥∥g− f−1∥∥
f(B(p,r)) < εr, ε < 1, Dg(f(p))−1 exists, and g(f(p)) = f−1 (f(p)) = p
From the first inequality in the above,
|g(f(x))−x|=∣∣g(f(x))− f−1 (f(x))
∣∣≤ ∥∥g− f−1∥∥f(B(p,r)) < εr
By Lemma 10.2.1,
g◦ f(
B(p,r))⊇ B(p,(1− ε)r) = B(g(f(p)) ,(1− ε)r)
Since Dg(f(p))−1 exists, it follows from the inverse function theorem that g−1 also existsand that g,g−1 are open maps on small open sets containing f(p) and p respectively. Thusthere exists η < (1− ε)r such that g−1 is an open map on B(p,η)⊆ B(p,(1− ε)r). Thus
g◦ f(
B(p,r))⊇ B(p,(1− ε)r)⊇ B(p,η)
So do g−1‘ to both ends. Then you have g−1 (p) = f(p) is in the open set g−1 (B(p,η)) .Thus
f(
B(p,r))⊇ g−1 (B(p,η))⊇ B
(g−1 (p) ,δ
)= B(f(p) ,δ )
pq◦ f
(B(p,r)
)B(p,(1− ε)r))
p = q(f(p))
With this lemma, the invariance of domain theorem comes right away. This remark-able theorem states that if f : U → Rn for U an open set in Rn and if f is one to one andcontinuous, then f(U) is also an open set in Rn.
Theorem 10.2.4 Let U be an open set in Rn and let f : U → Rn be one to one and contin-uous. Then f(U) is also an open subset in Rn.