Sivakumar M615-08c
Example Sheet 9
1. Suppose Xand Yare nonempty sets and f:X→Y.
(i) Show that ff−1(B)⊆Bfor every B⊆Y.
(ii) Show that A⊆f−1(f(A)) for every A⊆X.
2. Suppose (X, ρ) and (Y , ρ0) are metric spaces and f:X→Y. Prove that fis continuous on X
if and only if f(A)⊆f(A) for every A⊆X.
3. Suppose that (M, ρ) and (M0, ρ0) are metric spaces, and let fand gbe functions from Mto
M0. Assume that both fand gare continuous on M, and that f(x) = g(x) for every x∈D,
where Dis a dense subset of M. Show that f(x) = g(x) for every x∈M.
4. Suppose that (X, ρ) is a metric space, and let Tdenote the set {0,1}equipped with the discrete
metric. Prove that the following statements are equivalent:
(i) Xis connected.
(ii) If f:X→Tis continuous, then fis constant.
5. Give a direct proof of Theorem 3.8.9 (that is, show that given > 0, there is a δ > 0 such that
ρ0(f(s), f (t)) < whenever ρ(s, t)< δ).
6. Suppose (X, d) is a metric space.
(i) Let Bbe a nonempty subset of X. Define f:X→Ras follows:
f(x) := inf{d(x, b) : b∈B}, x ∈X.
(a) Show that f(x) = 0 if and only if xbelongs to the closure of B.
(b) Prove that
|f(x)−f(y)| ≤ d(x, y), x, y ∈X.
(c) Deduce that fis uniformly continuous on X(assuming that Ris equipped with the usual
metric).
(ii) Show that (X, d) is completely regular, to wit: given x0∈Xand a closed set Bwhich does
not contain x0, there is a continuous function g:X→R(where Ris equipped with the usual
metric) such that g(x0) = 1 and g(x) = 0 for every x∈B.
(iii) Deduce that (X, d) is regular, that is, given x0∈Xand a closed set Bwhich does not
contain x0, there exist d-open sets Uand Vsuch that x0∈U,B⊆V, and U∩V=∅.
7. Suppose that (X, d) is a metric space and let the real line Rbe equipped with the usual metric.
Let f:X→Rbe continuous on X. Show that the zero-set of f, namely,
Z(f) := {p∈X:f(p)=0}
is a closed subset of X.
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