Topology: Proving non-separability

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In summary: 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.
  • #1
tylerc1991
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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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
tylerc1991 said:
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.
 
  • #5
what is N, the natural numbers? what about N as a subset of itself

do you have a more formal definition for dense?
 
  • #6
tylerc1991 said:
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.
 
  • #7
Dick said:
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.
 
  • #8
tylerc1991 said:
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.
 

FAQ: Topology: Proving non-separability

What is topology?

Topology is a branch of mathematics that studies the properties of geometric shapes and spaces that are preserved under continuous deformations, such as stretching, bending, and twisting. It is concerned with the properties of objects that remain unchanged when they are stretched or squeezed, but not torn or glued together.

What is separability in topology?

Separability is a property of topological spaces that describes the ability to separate points from each other by open sets. A topological space is said to be separable if it contains a countable dense subset, meaning that there exists a countable set of points that are arbitrarily close to every point in the space.

How is non-separability proven in topology?

In order to prove that a topological space is non-separable, we need to show that it does not contain a countable dense subset. This can be done by constructing a set of points that are not arbitrarily close to any countable subset of the space, or by using other topological properties to show that a countable dense subset cannot exist.

Why is non-separability important in topology?

Non-separability is an important concept in topology because it helps us to understand the complexity of a topological space. It allows us to distinguish between different types of spaces and to classify them based on their properties. Non-separable spaces have unique topological properties that are not shared by separable spaces, which makes them interesting objects of study in mathematics and science.

What are some examples of non-separable spaces?

One example of a non-separable space is the real line with the Euclidean topology. Another example is the space of all continuous functions on a closed interval with the topology of pointwise convergence. Other examples include the Sorgenfrey line, the long line, and the Niemytzki plane.

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