Question

consider the set A=[0,1)U(1,2].
1) Find a finite open cover of A.
2) Show that A is not compact by using Heine-Borel theorem.
3) show that A is not compact by finding an open cover of A with no finite subcover.

Answer 1

1) \(\{\mathbb{R}\}\) certainly covers \(A\) and is a finite open cover.

Answer 2

Do you remember the Heine-Borel theorem?

Answer 3

It states that in \(\mathbb{R}\) a set is closed and bounded if and only if it is compact.

Answer 4

In other words, closed and boundedness is equivalent to compactness in \(\mathbb{R}\).

Answer 5

So what does that tell us about 2?

Answer 6

Yes, I do. #2 is not problem because it's obvious. the problem is #3

Answer 7

Ok

Answer 8

Is {R} finite or infinite?

Answer 9

A finite cover or finite subcover refers to the number of elements in the cover.

Answer 10

Let's discuss 3 and you will see what I mean.

Answer 11

The idea is to take a sequence of open sets that approaches 1. That way it is impossible to include only a finite number of sets in your subcover since you will not cover everything if you do.

Answer 12

\(\mathscr{C} = \{(-\infty, 1 - \frac{1}{n}), (1 + \frac{1}{n}, \infty) : n \in \mathbb{N}\}\)

Answer 13

I see. Could it be the union of intervals let say (-1,1-1/n)U(1+1/n, 3)?

Answer 14

Sure

Answer 15

You understand why it works?

Answer 16

The union of the elements of the cover certainly contains \(A\) so it is a valid cover.

Answer 17

However, it is impossible to take only a finite number of the elements to cover \(A\).

Answer 18

it works because intervals are close to 1 but don't include it. Am I right?

Answer 19

why not? we need to find finite open cover

Answer 20

Compactness states that every open cover has a finite subcover.

Answer 21

We have just found an open cover which does not have any finite subcover. Do you understand why there is no finite subcover?

Answer 22

no yet

Answer 23

Can you find a finite subset of \(\mathscr{C}\) which covers \(A\)? If not, why?

Answer 24

Wait. we need to show that A is not compact by finding open cover with no finite subcovers. if there are finite subcovers, the set is compact. Yes?

Answer 25

Compactness states that every open cover has a finite subcover. So to prove that a set is compact you must show that every possible open cover has a finite subcover. On the other hand to show a set is NOT compact, you must only provide one open cover that has no finite subcover.

Answer 26

Once again we return to our solution:
\(\mathscr{C} = \{(-\infty, 1 - \frac{1}{n}), (1 + \frac{1}{n}, \infty) : n \in \mathbb{N}\}\)

We claim it is an open cover, but it has no finite subcover. If this is true then \(A\) is not compact.

Answer 27

Let is carefully define what the terms mean to avoid confusion. An open cover for some set \(A\) is a collection of open sets, \(\mathscr{C}\), whose union contains \(A\). A finite subcover of \(\mathscr{C}\) is a finite subset \(C \subset \mathscr{C}\) such that \(C\) is still a cover for \(A\). In other words it is a finite subset of \(\mathscr{C}\) whose union contains \(A\). Do you understand the definitions of these terms?

Answer 28

Yes, but we need to find in #1 open finite cover. i guess it is some open interval which differs from (-inf, inf)

Answer 29

1) Is a bit strange and rather trivial. You usually don't talk about finite open covers. An open cover of some set \(A\) is simply any collection of open sets whose union contains \(A\). A finite open cover is simply a finite collection meeting the above requirements.

Answer 30

So let me ask you, is \(\mathscr{C} = \{\mathbb{R}\}\) an open cover for \(A\)?

Answer 31

it's open, but not finite

Answer 32

Finite just means the cover has a finite number of elements. Tell me, how many elements does \(\mathscr{C}\) have?

Answer 33

Do you mean just one?

Answer 34

Correct.

Answer 35

If you are troubled by including the entire space, another solution would be \(\mathscr{C} = \{(-1, 3)\}\). But both are valid solutions.

Answer 36

In any case, I think question #1 is simply there to test your understanding of the definitions. It is trivial.

Answer 37

Do you understand the solution for 3?

Answer 38

I think I do. I just need to reread some theoretical stuff.
Thank you so much for help and clarifying.

Answer 39

If you have any questions about definitions, you can certainly continue to ask here. Try to carefully understand what compactness means. To understand compactness you also need to understand what an open cover is and what a finite subcover is.