Is Every Infinite Set Countable? Understanding Finite vs. Infinite in Topology

[rumjum]

Homework Statement

I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable.

1)So, anything that is finitie is countable. Anything that is infinite is also countable?
Then what is uncountable, something that is both uncountable and infinite.

2) It is mentioned that line [a,b] is uncountable. But, why?

3) Also if a set is uncountable then the complement of that set is countable? I don't think so because for all x that belong to R and do not belong to [a,b]. The set still belongs to R and should be uncountable.


Can someone explain these loose ends of my understanding?

Homework Equations

The Attempt at a Solution

 

[ZioX]

Uncountable means that you can't count EVERY element in it. Uncountable immediately implies infinite, as every finite set is countable (count them!).

My new catchphrase: If only there was some global database available to the public to freely share information.

 

[Dick]

Uncountable means that you can't count EVERY element in it. Uncountable immediately implies infinite, as every finite set is countable (count them!).

My new catchphrase: If only there was some global database available to the public to freely share information.

By count, ZioX means index with integers. There are sets you can't do this with. See 'Cantor diagonal arguments'. A set could be uncountable and have a countable complement. It also could not. Give examples of both, please??

 

[HallsofIvy]

So, anything that is finitie is countable. Anything that is infinite is also countable?

No, for example, the set of all real numbers is uncountable.

Then what is uncountable, something that is both uncountable and infinite.

Yes, but normally we don't bother to say "infinite" since all finite sets ARE countable. "Uncountable" itself implies "infinite".

2) It is mentioned that line [a,b] is uncountable. But, why?

As said before, Cantor's diagonal argument. You should have see that when you were first introduced to "countable" and "uncountable".

3) Also if a set is uncountable then the complement of that set is countable? I don't think so because for all x that belong to R and do not belong to [a,b]. The set still belongs to R and should be uncountable.

Certainly not! Why would you think so? Yes, the set of all real numbers between 0 and 2 is uncountable. The set of all real numbers between 0 and 1 is uncountable. It's complement in the set of all real numbers between 0 and 2 is the set of all real numbers between 1 and 2 which is still uncountable.

Can someone explain these loose ends of my understanding?

Homework Equations

The Attempt at a Solution

 

[CompuChip]

I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable.

If that set is 1, 2, 3, 4, ... (or a subset of it) - then yes.
For example, the set of numbers {1, 2, 3} is clearly countable as I can count all the numbers in them. So is the set {1, 2, 3, 4, ...} (all natural numbers) - I can assign each number to itself. The set {2, 4, 6, 8, ...} of even numbers is countable, because I can assign to each number n the number 2n (that is, describe the set as a_n = 2n). Also, the set of all integers is countable (just write them in the peculiar order {0, 1, -1, 2, -2, 3, -3, 4, -4, ... } and you see that you can list them, in principle).

The set of all real numbers between 0 and 1 is not countable. For any countable list you give me, I can always give you one that is not in the list (this is the Cantor diagonal argument mentioned). Also, the set of continuous functions on [0, 1] with f(0) = 0 is uncountable, for example, the set of functions
\{ f(x) = a x \mid a \in \mathbb{R} \}
is uncountable (even if we restrict a to the interval [0, 1], it is basically the same as this interval since each choice for a will define such a function). But then the subset of all functions
\{ \sin 2 \pi n x \mid n \in \mathbb{N} \}
is countable, as each number n fixes such a function.