Topology: Proving non-separability

[tylerc1991]

Homework Statement

Show that any countable subset of N with the discrete topology cannot be dense in N.

Homework Equations

Informally a set is dense if, for every point in X, the point is either in A or arbitrarily "close" to a member of A

The Attempt at a Solution

I was thinking of using the prime numbers as my subset since each prime number is not necessarily arbitrarily close to another prime number due to the sporadic nature of the primes. Any help/input would be greatly appreciated.

 

[Dick]

N is the integers? With the discrete topology? You are way overthinking this. There is no proper subset of N that is dense in N.

 

[tylerc1991]

Sorry, N is the natural numbers. This sub-problem stems from the original problem: Give an example of a separable hausdorff space with a subspace that is not separable.

So what I decided on was this: letting the separable hausdorff space be the natural numbers with the discrete topology and letting the subspace be the answer i gave originally.

 

[Dick]

Sorry, N is the natural numbers. This sub-problem stems from the original problem: Give an example of a separable hausdorff space with a subspace that is not separable.

So what I decided on was this: letting the separable hausdorff space be the natural numbers with the discrete topology and letting the subspace be the answer i gave originally.

Then your example is way off. If X is ANY space with the discrete topology then any proper subset of X isn't dense. Better think of another example. Discrete doesn't work.

 

[lanedance]

what is N, the natural numbers? what about N as a subset of itself

do you have a more formal definition for dense?

 

[Dick]

Sorry, N is the natural numbers. This sub-problem stems from the original problem: Give an example of a separable hausdorff space with a subspace that is not separable.

So what I decided on was this: letting the separable hausdorff space be the natural numbers with the discrete topology and letting the subspace be the answer i gave originally.

Examples of a separable space with a subspace that's not separable are fairly exotic. Maybe you want to think of this as literature search project rather than just making one up. I hate to say this, but Google it.

 

[tylerc1991]

Examples of a separable space with a subspace that's not separable are fairly exotic. Maybe you want to think of this as literature search project rather than just making one up. I hate to say this, but Google it.

I will see the prof. tomorrow. I agree, I have been Googling this question for some time and the answers I have been getting are quite foreign looking. Because of this I think the prof. will give me some leniency.

 

[Dick]

I will see the prof. tomorrow. I agree, I have been Googling this question for some time and the answers I have been getting are quite foreign looking. Because of this I think the prof. will give me some leniency.

Did you find the Moore plane? I did. It's the sort of thing you looking for but I doubt I'd have been able to make something like that up for a homework exercise.