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1. (a) Let Q be the set of rational numbers. Show that the Riemann integral of 1Q from 0 to 1 is undefined (the net in its definition does not converge). (Q is countable and [0, 1] is uncountable, so the integral "should be" 0, and will be for the Lebesgue integral, to be defined in Chapter 3.)
(b) Show that for a sequence 1F (n) of indicator functions of finite sets F (n) converging pointwise to 1Q, the Riemann integral of 1F (n) is 0 for each n.
2. Let X be an infinite set. Let T consist of the empty set and all complements of finite subsets of X . Show that T is a topology in which every singleton {x} is closed, but T is not metrizable. Hint: A sequence of distinct points converges to every point.
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