15.2. OPEN AND CLOSED SETS 311

be in C. This is a rough heuristic explanation of what is going on with these definitions.Also note that Rp and /0 are both open and closed. Here is why. If x ∈ /0, then there mustbe a ball centered at x which is also contained in /0. This must be considered to be truebecause there is nothing in /0 so there can be no example to show it false1. Therefore, fromthe definition, it follows /0 is open. It is also closed because if x /∈ /0, then B(x,1) is alsocontained in Rp \ /0 =Rp. Therefore, /0 is both open and closed. From this, it follows Rp isalso both open and closed.

Theorem 15.2.3 Let x ∈ Rp and let r ≥ 0. Then B(x,r) is an open set. Also,D(x,r) ≡ {y ∈ Rp : |y−x| ≤ r} is a closed set. In particular, every closed interval in Ris a closed set.

Proof: Suppose y ∈ B(x,r). It is necessary to show there exists r1 > 0 such thatB(y,r1)⊆ B(x,r). Define r1 ≡ r−|x−y|. Then if |z−y|< r1, it follows from the abovetriangle inequality that

|z−x| = |z−y+y−x| ≤ |z−y|+ |y−x|< r1 + |y−x|= r−|x−y|+ |y−x|= r.

Note that if r = 0 then B(x,r) = /0, the empty set. This is because if y ∈ Rp, |x−y| ≥ 0and so y /∈ B(x,0). Since /0 has no points in it, it must be open because every point in it,(There are none.) satisfies the desired property of being an interior point.

Now suppose y /∈ D(x,r). Then |x−y| > r and defining δ ≡ |x−y| − r, it followsthat if z ∈ B(y,δ ), then by the triangle inequality,

|x−z| ≥ |x−y|− |y−z|> |x−y|−δ

= |x−y|− (|x−y|− r) = r

and this shows that B(y,δ )⊆Rp \D(x,r). Since y was an arbitrary point in Rp \D(x,r),it follows Rp \D(x,r) is an open set which shows, from the definition, that D(x,r) is aclosed set as claimed. Now [a,b] = D

( a+b2 , b−a

2

).

A picture which is descriptive of the conclusion of the above theorem which also im-plies the manner of proof is the following.

1To a mathematician, the statement: Whenever a pig is born with wings it can fly must be taken as true. Wedo not consider biological or aerodynamic considerations in such statements. There is no such thing as a wingedpig and therefore, all winged pigs must be superb flyers since there can be no example of one which is not. Onthe other hand we would also consider the statement: Whenever a pig is born with wings it cannot possibly fly,as equally true. The point is, you can say anything you want about the elements of the empty set and no one cangainsay your statement. Therefore, such statements are considered as true by default. You may say this is a verystrange way of thinking about truth and ultimately this is because mathematics is not about truth. It is more aboutconsistency and logic.