Exercise 2.14

Exercise 2.14: Find an example to show that a union of infinitely many closed sets may not be closed.

Olomana
First, note that closed sets are defined as the complements of open sets. By De Morgan's Law, this exercise is equivalent to finding an infinite intersection of open sets which is not open, which we've already done for metric spaces in Exercise 1.12.b. Since metric spaces induce topological spaces, this provides an example for topological spaces.

However, let's construct an example without using a metric.

Consider the cofinite topology $$\mathcal{T}$$ on any infinite set $$X$$. Let $$x$$ be any element of $$X$$, and let $$U$$ be the complement of the singleton set $$\{x\}$$. Since $$\{x\}$$ is finite, it is not in $$\mathcal{T}$$ and is therefore not open. Since $$\{x\}$$ is not open, its complement $$U$$ is not closed.

Note that $$U$$ is an infinite set. Consider the singleton sets $$\{u_i\} : u_i \in U$$. The infinite union of these singleton sets is $$U$$ itself. All singleton sets are finite, and are closed because they're the complements of open sets. Thus we have an example of an infinite union of closed sets that is not closed.

Finally, note that both $$X$$ and $$\emptyset$$ are open and closed, so we constructed $$U$$ with 'every element of $$X$$ except one' to get a set that is open and not closed, with a complement that is closed and not open.

Olomana 07:50, March 15, 2011 (UTC)

'''Cool. Linked a couple things and changed "metric spaces are topological spaces" to "metric spaces induce topological spaces". --Steven.clontz 20:11, March 17, 2011 (UTC)'''