Class Contents

• Closed sets

Closed Sets

Example:

Example:

Any closed interval [a,b] is closed.

Proof.

Example:

Example:

Any finite set is closed.

Proof.

Proof.

Example:

Proof.

Example:

Remark:

Notice the versatility of our unioning notation.

The set K is quite a strange set. Notice that although it contains

it does not contain any intervals.

This set is called the Cantor Middle 1/3 Set. We'll come back to it later on.

Remark:

Remember that the union of a collection of open sets is also open. The corresponding result for closed sets is:

I want to show you two ways of proving this. One, directly, and the other, by deriving it from what we already know about open sets.

First Proof.

To see another, quicker way of proving the last Proposition, we use the following factoid:

Factoid:

Proof of factoid.

Thus, we can use Proposition 3.15 by to prove Proposition 3.16 as folllows:

Proof.

Example:

Proof.

Example:

Example:

Proof.

Example:

Example:

The proof is in the homework or later.

John, do you want the above statement to stand, or do you want to write a proof for it?

Remark:

Check out the animated demonstration of the construction of the Cantor Set

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: May 1996