230 CHAPTER 12. VECTOR VALUED FUNCTIONS

To illustrate this definition, consider the following picture.

x UB(x,r)

You see in this picture how the edges are dotted. This is because an open set, can not in-clude the edges or the set would fail to be open. For example, consider what would happenif you picked a point out on the edge of U in the above picture. Every open ball centeredat that point would have in it some points which are outside U . Therefore, such a pointwould violate the above definition. You also see the edges of B(x,r) dotted suggesting thatB(x,r) ought to be an open set. This is intuitively clear but does require a proof. This willbe done in the next theorem and will give examples of open sets. Also, you can see that ifx 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 do not have their skins while closed sets do. Hereis a picture of a closed set, C.

B(x,r)xC

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

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.