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Class Contents

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.

Remark: