Closed sets in a metric space, Real Analysis II
In this lecture, we define the notion of a closed subset of a metric space: a set is closed if its complement is open. Thus closed and open are related concepts but not opposites (like for windows and doors!). We then work through several examples to illustrate this definition.
(MA 426 Real Analysis II, Lecture 8)
First, we discuss that in any metric space, both the entire space and the empty set are always closed. Next, we show that the interval [0,1] is indeed closed because its complement, which is the union of two open intervals, is open. We also consider a set in R^2 consisting of all points whose distance to the origin is less than or equal to 10, and we demonstrate that this set is closed because its complement is open.
In the fourth example, we examine the set of rational numbers as a subset of the real line and explain that neither the rationals nor the irrationals are closed since their complements are not open.
We then explore three additional examples, starting with the set of integers, which is closed because its complement is a union of open intervals. We discuss the Archimedean set, which consists of points of the form 1/n, and explain that it is not closed because its complement is not open. Finally, we address subsets of a discrete metric space, noting that in such spaces, every set is both open and closed.
We conclude the lecture by proving a theorem that states three key properties of closed sets: the entire metric space and the empty set are always closed, any intersection of closed sets is closed, and any finite union of closed sets is closed. We also construct counterexamples to show that an infinite union of closed sets may not be closed.
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