## 5.1. Open and Closed Sets | IRA |

All of the previous sections were, in effect, based on the natural numbers. Those numbers were postulated as existing and all other properties - including other number systems - were deduced from those numbers and a few principles of logic.

We will now proceed in a similar way: first, we need to define the basic objects we want to deal with, together with their most elementary properties. Then we will develop a theory of those objects and called it topology.

Definition 5.1.1: Open and Closed SetsA set URis calledopen, if for each xUthere exists and > 0 such that the interval ( x - , x + ) is contained inU. Such an interval is often called an- neighborhoodof x, or simply a neighborhood of x.A set

Fis calledclosedif the complement ofF,R\F, is open.

It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets. What about combining infinitely many sets ?

Examples 5.1.2:

Proposition 5.1.3: Unions of Open Sets, Intersections of Closed Sets

- Every union of open sets is again open.
- Every intersection of closed sets is again closed.
- Every finite intersection of open sets is again open
- Every finite union of closed sets is again closed.

How complicated can an open or closed set really be ? The basic open (or closed) sets in the real line are the intervals, and they are certainly not complicated. As it will turn out, open sets in the real line are generally easy, while closed sets can be very complicated.

The worst-case scenario for the open sets, in fact, will be given in the next result, and we will concentrate on closed sets for much of the rest of this chapter.

Next we need to establish some relationship between topology and our previous studies, in particular sequences of real numbers. We shall need the following definitions:

Proposition 5.1.4: Characterizing Open SetsLet URbe an arbitrary open set. Then there are countably many pairwise disjoint open intervals U_{n}such thatU= U_{n}

Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated PointsLet Sbe an arbitrary set in the real lineR.

- A point b
Ris calledboundary pointofSif every non-empty neighborhood of b intersectsSand the complement ofS. The set of all boundary points ofSis called the boundary ofS, denoted bybd(S).- A point s
Sis calledinterior pointofSif there exists a neighborhood ofScompletely contained inS. The set of all interior points ofSis called the interior, denoted byint(S).- A point t
Sis calledisolated pointofSif there exists a neighborhoodUof t such thatUS= {t}.- A point r
Sis calledaccumulation point, if every neighborhood of r contains infinitely many distinct points ofS.

Here are some results that relate these various definitions with each other.

Examples 5.1.6:

Finally, here is a theorem that relates these topological concepts with our previous notion of sequences.

Proposition 5.1.7: Boundary, Accumulation, Interior, and Isolated Points

- Let
SR. Then each point ofSis either an interior point or a boundary point.- Let
SR. Thenbd(S) =bd(R\S).- A closed set contains all of its boundary points. An open set contains none of its boundary points.
- Every non-isolated boundary point of a set
SRis an accumulation point ofS.- An accumulation point is never an isolated point.

Theorem 5.1.8: Closed Sets, Accumulation Points, and Sequences

- A set
SRis closed if and only if every Cauchy sequence of elements inShas a limit that is contained inS.- Every bounded, infinite subset of
Rhas an accumulation point.- If
Sis closed and bounded, and is any sequence inS, then there exists a subsequence of that converges to an element ofS.