# How do I show that the real numbers are not compact?

• fred123
In summary, the conversation discusses a question about proving that the real numbers are not compact using only the definition of compactness. The participants suggest using a counterexample and finding an open cover that cannot be reduced to a finite subcover. Possible solutions are proposed, including using the set { (n-1, n+1) | n in Z } or the family of sets S={ (-n,n) | n in N }. The conversation also touches on the importance of properly defining open covers and the use of logical statements in solving problems.
fred123
A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R?

I let V be a collection of open subset that cover R
Then I make the assumption that that this open cover can be reduced to a finite subcover.
(Clearly this is not possible) I am struggling to see/show what happens to R when I make this assumption. Should I simply find a counterexample, if so, what could it look like?

I have proven several, similar problems, but this one is so general that its tricky.

Last edited:
You have to check two properties:
Closed?
Bounded?
What do you think?

But the question states that I should do this proof directly from the definition of being compact. Ie, if an open cover can be reduced to a finite subcover.

Compact: every open cover has a finite subcover.
So what is the definition of non-compact? Now show the reals satisfy this definition. If you can negate logical statements then this problem is not at all tricky.

So, find one open cover that cannot be reduced to a finite subcover?
But wouldn't that imply that I have to state what this open cover looks like? Should I maybe let R be an open cover of itself?

That open cover has exactly one set in it, R. One is a finite number.

Find an open cover of infinitely many sets that does not have a finite subcover. Post what you're thinking, i.e. how you might make such a cover.

Please, no one post a solution right away.

I think I sorted some stuff out, and {n-1,n+1 n in Z} I believe must work!? Thanks for the support

Well, I would write it (n-1,n+1) for n contained in Z. {n-1, n+1} means the set containing exactly two members, n-1 and n+1. That's not open. (I don't know what "n in Z" inside the braces could mean!)

You might try and have a closer look at the family of sets $S=\{(-n,n),n\in\mathbb{N}\}$

Fred's works too (and was essentially my first thought), assuming he meant { (n-1, n+1) | n in Z }.

Agreed. I was unsure of what he meant with the type of parentheses he used.

## What is the definition of compactness for real numbers?

The definition of compactness for real numbers is that every open cover has a finite subcover. In other words, if a set of real numbers can be covered by a finite number of open intervals, then it is considered compact.

## How do I prove that the real numbers are not compact?

To prove that the real numbers are not compact, you can use a counterexample. One commonly used counterexample is the set of real numbers between 0 and 1, which can be covered by infinitely many open intervals but does not have a finite subcover.

## Can you explain why the real numbers are not compact?

The real numbers are not compact because they do not satisfy the definition of compactness. There exists at least one open cover of the real numbers that cannot be reduced to a finite subcover, which contradicts the definition of compactness.

## Are there any other proofs of the non-compactness of the real numbers?

Yes, there are various other ways to prove that the real numbers are not compact. Some involve using the properties of the real numbers, such as the Archimedean property, while others use concepts from topology, such as the Bolzano-Weierstrass theorem.

## How is the non-compactness of the real numbers relevant in mathematics?

The non-compactness of the real numbers has important implications in various areas of mathematics, such as analysis and topology. It also serves as a counterexample in many proofs and helps to distinguish between compact and non-compact spaces.

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