11.2 Continuity and Compactness
k 1£(TE).
1/2
= 6.Proposition 11.1.9 / is continuous if and only if every component is con-
tinuous.Proof. (=>•): Fix j. Show that fj is continuous: Fix p £ X. Show that fj
is continuous at p. Given e > 0 35 > 0 3 Vx with d 2 {x,p) < 5, then
1/iW - /»l = *(/>(*),/;(p)) < d 2 (f(x),f(p)) < e.
(<=): Assume that Vj, fj is continuous atpSl. Show that / is continuous
at p. Let e > 0 be given.
/i is continuous at p => 35\ > 0 3 d 2 (x,p) < S\ =>• |/i(x) — /i(p)| < -4?.
/ 2 is continuous at p => 3<5 2 > 0 9 d 2 (x,p) < 52 => 1/2(2;) — /2(f)! < TTJ-/it is continuous at p => 34 > 0 3 d 2 (x,p) < 5k => |/fc(a;) - /fe(p)| < -j%-
Let 5 = min{#i,...,5k} > 0. Let X be 9 d(x,p) < 5. Then,d2(f(x),f(p)) = [J2\fi^)-fi(p)\
2
]
1/2
<11.4 Continuity and Connectedness
Theorem 11.2.1 The continuous image of a compact space is compact, i.e.
if f : X i-t Y is continuous and (X,d) is compact, then f(X) is a compact
subspace of (Y, dy).Proof. Let {Va : a 6 A} be any open cover of f(X). Since / is continuous,
f-l(Va) is open in X. f(x) C \Ja&A Va =» X C /" H/fr)) C \JaeA f~HVa).
Since X is compact, 3au...,an 9 X C [/_1(VQl)|J-' 'U/"^1 ^™)] =>
/(*) C /[/-^1 (Val)U---U/_1(^aJ] = ^U-'-U^, since for A C
/-'/W.rV^CBwehave/(IK) = U/(^)
and
/_1
(IK) = U/"1
^)-D
Corollary 11.2.2 yl continuous real valued function on a compact metric
space attains its maximum and minimum.Proof. f(X) is a compact subset of K => f(X) is bounded. Let m =
inf/(a;), M = sup f(x). Then, m,M e R; since f(X) is bounded. Also,
m,M e f(X). Furthermore, f(x) = f(x), since /(X) is compact. Thus,
3p e X 3 m = /(p) and 3? 6 I 3 M = /(g). Finally, m = /(p) < f{x) <
f{q) = M,VxeX. D
Theorem 11.2.3 Let (X,dx) be a compact metric space, (Y, dy) be a metric
space, f : X H-» Y be continuous, one-to-one and onto. Then, f~x :Y>-¥Xis
continuous.