98 CHAPTER 6. MULTI-VARIABLE CALCULUS

that B(x,r) ought to be an open set. This is intuitively clear but does require a proof. Thiswill be done in the next theorem and will give examples of open sets. Also, you can seethat if x is close to the edge of U, you might have to take r to be very small.

It is roughly the case that open sets don’t have their skins while closed sets do. Here isa picture of a closed set, C.

B(x,r)xC

Note that x /∈C and since Fn \C is open, there exists a ball, B(x,r) contained entirelyin Fn \C. If you look at Fn \C, what would be its skin? It can’t be in Fn \C and so it mustbe in C. This is a rough heuristic explanation of what is going on with these definitions.Also note that Fn 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 Fn \ /0 = Fn. Therefore, /0 is both open and closed. From this, it follows Fn isalso both open and closed.

Theorem 6.2.3 Let x ∈ Fn and let r ≥ 0. Then B(x,r) is an open set. Also,

D(x,r)≡ {y ∈ Fn : |y−x| ≤ r}

is 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.

1To a mathematician, the statment: 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 can’t possibly fly, asequally 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.