8  Properties of Open Sets

Highlights

• Characterization of open sets as unions of open balls
• Unions and intersections of open sets

8.1 Preparation

We’ll start with a few warm-up exercises using the definition of an open set from Chapter 7.

Exercise 8.1  

1. Write down what it means for a subset of a metric space to not be open.
2. Write down a careful proof that \((-\infty,2]\) is not an open subset of \(\R\).

Caution

We are not going to call non-open sets “closed”. The term “closed” will have a different definition that we’ll get to later.

Exercise 8.2 Let \(U\) be an open subset of \(\R\). Prove that \(U\) has no greatest element.

The next theorem gives us another useful characterization of open sets—that each is a union of a collection of open balls.

Theorem 8.3 Let \(X\) be a metric space and \(U\subseteq X\). The following are equivalent.

1. \(U\) is open.
2. \(U\) is a union of open balls.

Slide to reveal

Desmos Applet 8.1: A demonstration of the second condition of Theorem 8.3. The open set \(S\) is union of open balls.

Theorem 8.3 is another “the following are equivalent” theorem. We love these because anytime there is a proof involving an open set, we get to pick which condition to use. Before getting into the proof of this theorem, we’ll see how it can be used.

Exercise 8.4 Redo Exercise 7.13 using Condition 2 of Theorem 8.3 by showing that the interval \((-\infty,3)\) is a union of open balls.

8.2 Open Sets as Unions of Open Balls

Now we get to the proof.

Exercise 8.5 Prove Theorem 8.3.

It is worth mentioning that some open sets are easier to witness as unions of open balls than others. Compare the two examples below.

Exercise 8.6 Prove that each set below is open. Think about which condition of Theorem 8.3 will be easiest to use in each case.

1. The complement of \(\Z\) in \(\R\).
2. The upper half plane \(H=\{(x,y)\in\R^2\st y>0\}\) as a subset of \(\R^2\).

Number of open balls:

Desmos Applet 8.2: The upper half plane from Exercise 8.6 is hard to imagine as a union of open balls since finitely many cannot cover the plane.

8.3 Unions and Intersections of Open Sets

We can construct new open sets out of old ones using unions and intersections via the following theorem.

Theorem 8.7  

1. The union of any collection of open sets is open.
2. The intersection of any finite collection of open sets is open.

Exercise 8.8 Prove Part 1 of Theorem 8.7.

Hint: Will it be easier to work with the definition of openness, or Condition 2 of Theorem 8.3 for this proof?

Proof (Setup of proof of Part 2 of Theorem 8.7). Suppose \(U_1,\ldots,U_n\) are open and let \(U=\bigcap_{i=1}^n U_i\). We will prove \(U\) is open using the definition of openness, so we start by supposing that \(x\in U\). This means \(x\in U_i\) for all \(i=1,\ldots,n\). Therefore, applying the definition of openness to each \(U_i\), there exists \(r_i\) such that \(\bee_{r_i}(x)\subseteq U_i\).

Show \(B_{r_1}\)

Show \(B_{r_2}\)

Desmos Applet 8.3: Here is an illustration of what’s going on in the proof so far with just \(n=2\) open sets. Here, \(x\) is in \(U_1\), so there is an open ball around \(x\) which lies entirely within \(U_1\). Similarly, \(x\) is in \(U_2\), so there is an open ball around \(x\) which lies entirely within \(U_2\). What of these two open balls could you use to finish proving that the intersection is open? How would you generalize this to the case where there are \(n\) open sets?

Exercise 8.9 Finish the proof of Part 2 of Theorem 8.7.

Exercise 8.10  

1. At what point in the proof of Part 2 of Theorem 8.7 did you use the fact that we were only dealing with finitely many open sets?
2. Give an example of a(n infinite) collection of open sets whose intersection is not open. (Remember, \(\varnothing\) is open!)

Review Questions

1. Say what it means for a set to be “not open”.
2. What are the equivalent characterizations of open sets?
3. How do open sets behave under unions and intersections?