next up previous
Next: About this document ...


MAT425 Real Analysis L11

Prof. Thistleton



More (Very) Basic Topology We generalize the idea of a closed interval.

Definition 139

Let \( F \subseteq \Re \). We say that F is closed if it contains each of its cluster points.

Examples


























Theorem 140

The union of two closed sets is closed.


Theorem 141

The union of a finite number of closed sets is closed.




















Theorem 142

The intersection of two closed sets is closed.




















Theorem 143

The intersection of any collection of closed sets is closed.


Theorem 144

The complement of a closed set is open.


























Theorem 145

The complement of an open set is closed.


Theorem 151

A set \( S \neq \emptyset \) is open if and only if it is the union of a countable collection of mutually disjoint open intervals. That is, \( S = \cup ^{\infty}_{n=1} U_n \) where each Ui is an open interval and \( U_i \cap U_j = \emptyset \) for each \(i \neq j\).

Discussion This proof is an important one because it allows us to characterize any generic open set in terms of simple building blocks, open intervals. We can not do this for closed sets. The proof will be organized along the following lines:

1.
Characterize S as the union of a collection of open intervals, but as yet with no apparent structure.

2.
Show that the open intervals making up this collection must be disjoint or identical.

3.
Show that the collection of open intervals is, in fact, countable.


The Proof


1.
\( \forall x \in S\) define \(A_x = \{ a\ \vert\ a<x\ and\ (a,x] \subseteq S \}\) and define \(B_x = \{ b\ \vert\ x<b\ and\ [x,b) \subseteq S \}\)




(a)
If Ax is bounded below define ax = inf(Ax), otherwise let \( a_x = -\infty\).
(b)
If Bx is bounded above, then define bx = sup(Bx), otherwise \( b_x = +\infty\).




Now show that \( S = \cup_{x \in S} (a_x,b_x) \)


2.
In order to show that the intervals making up this collection are disjoint or identical note that \( a_x \notin S\) for all x. Also, show that \( b_x \notin S\) for all x.
































Since the ax and the bx are not in S we have that any two of these intervals are disjoint or identical.

3.
Now form a one to one correspondence between the above intervals and the rationals.

 
next up previous
Next: About this document ...
William Thistleton
11/10/1998