Math 524
Homework due 10/25/06
Reading
Section 10 Chapter 7 in Royden.
Sections 1, 2, 3 and 4 Chapter 3 in Royden.
Sections 1, 2, 3, 4 and 5 Chapter 1 in Folland.
Problem from Royden:
Chapter 7, Section 10: problem 50.
1. (a) Let Xbe a metric space; let Obe an open subset of X. Prove that the points of
discontinuity of the characteristic function of O,χO, form a nowhere dense subset
of X.
(b) Assume that Xis a complete metric space, and {Ui}∞
i=1 any countable collection
of open sets. Show that there exists x∈Xsuch that χUiis continuous at xfor
each i.
2. Let Xbe a metric space. Prove that Xis complete if and only if for any decreasing
sequence of non-empty closed subsets of X,{An}, (i.e.· · · ⊂ An⊂An−1· · · ⊂ A2⊂A1)
such that
lim
n→∞ diam An= 0,
∩∞
n=1An={x}for some x∈X.
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