310 CHAPTER 15. SEQUENCES, COMPACTNESS, CONTINUITY

15.2 Open and Closed SetsOpen sets are those sets S such that if x ∈ S, then so is y whenever y is sufficiently close tox. Closed sets are those sets S such that if xn → x and each xn ∈ S, then also x ∈ S. Whatfollows is just a more precise statement of this.

Eventually, one must consider functions which are defined on subsets of Rp and theirproperties. The next definition will end up being quite important. It describe a type ofsubset of Rp with the property that if x is in this set, then so is y whenever y is closeenough to x.

Definition 15.2.1 Recall for x,y ∈ Rp, |x−y| =(

∑pi=1 |xi − yi|2

)1/2. Also let

B(x,r)≡ {y ∈ Rp : |x−y|< r} . Let U ⊆ Rp. U is an open set if whenever x ∈U, thereexists r > 0 such that B(x,r) ⊆ U. More generally, if U is any subset of Rp, x ∈ U is aninterior point of U if there exists r > 0 such that x ∈ B(x,r)⊆U. In other words U is anopen set exactly when every point of U is an interior point of U.

If there is something called an open set, surely there should be something called aclosed set and here is the definition of one.

Definition 15.2.2 A subset, C, of Rp is called a closed set if Rp \C is an openset. They symbol Rp \C denotes everything in Rp which is not in C. It is also called thecomplement of C. The symbol SC is a short way of writing Rp \S.

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 thatif x is close to the edge of U , you might have to take r to be very small. open sets do nothave their skins while closed sets do. Here is a picture of a closed set, C.

B(x,r)xC

Note that x /∈C and since Rp \C is open, there exists a ball, B(x,r) contained entirelyin Rp \C. If you look at Rp \C, what would be its skin? It can’t be in Rp \C and so it must