Is the Natural Numbers Dense in Itself?

[Bachelier]

$$\ Is \ \mathbb{N} \ dense \ in \ itself.$$

 

[lurflurf]

Yes, every space is dense in itself.

 

[HallsofIvy]

"A is dense in B" (with A and B topological spaces) mean "given any point p in B, every open set containing p contains some point of A." Of course, if A= B, that is trivially true.

 

[mjpam]

"A is dense in B" (with A and B topological spaces) mean "given any point p in B, every open set containing p contains some point of A." Of course, if A= B, that is trivially true.

So is there a point $p\in(n,n+1)\forall n\in\mathbb{N}$ such that $p\in\mathbb{N}$?

 

[micromass]

So is there a point $p\in(n,n+1)\forall n\in\mathbb{N}$ such that $p\in\mathbb{N}$?

No, but that doesn't matter. We're talking about denseness of N in N. Your example doesn't apply because you're confused with showing that N is dense in R!Also, for the OP, note that there are different (non-equivalent) definitions of denseness. Most often dense is applied in topological spaces, and this is what people in this thread do. But there are other definitions of denseness such that N is not dense in N. I'm just saying this because this is probably what confuses you. But you should always check what definition of denseness you are using.

 

[mjpam]

No, but that doesn't matter. We're talking about denseness of N in N. Your example doesn't apply because you're confused with showing that N is dense in R!Also, for the OP, note that there are different (non-equivalent) definitions of denseness. Most often dense is applied in topological spaces, and this is what people in this thread do. But there are other definitions of denseness such that N is not dense in N. I'm just saying this because this is probably what confuses you. But you should always check what definition of denseness you are using.

No, I just misread the original statement to say that "every open subset of A must contains a member of A". After re-reading the original statement I see how, by that definition of "dense", every set is trivially dense in itself.

 

[Landau]

Also, for the OP, note that there are different (non-equivalent) definitions of denseness. Most often dense is applied in topological spaces, and this is what people in this thread do. But there are other definitions of denseness such that N is not dense in N. I'm just saying this because this is probably what confuses you. But you should always check what definition of denseness you are using.

What kind of 'denseness' do you have in mind here? Some measure theoretic concept?

 

[micromass]

No, there are some order-theoretic notions of denseness. For example:

for every x and y, there exists a z such that x<z<y. This is sometimes called denseness. I just mention this, because the OP has talked about this in another post. I just wanted to clear up what definition of denseness we're using here...

 

[Landau]

Right. So the order on N is definitely not dense.

And then there's the term dense-in-itself, meaning 'containing no isolated points' (which of course depends on the superset you're considering.)