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## Main Question or Discussion Point

I can add 2 dense sets together and get a non dense set right?

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I can add 2 dense sets together and get a non dense set right?

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HallsofIvy

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tiny-tim

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hi cragar!

eg the Cantor set is uncountable but nowhere dense …

nopeA dense set means its uncountable right?

eg the Cantor set is uncountable but nowhere dense …

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And uncountable meaning If im at one number I couldn't tell you the next number in the list.

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tiny-tim

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depends what you mean by "list"And uncountable meaning If im at one number I couldn't tell you the next number in the list.

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tiny-tim

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for any number x in [0,1), i make a sub-list {… x-3 x-2 x-1 x x+1 x+2 …}, and i order all the sub-lists in order of x …

that way, there is *always* a next number!

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No, that's not at all what it means! Uncountable simply means that there is no injection from the set to the natural numbers.

The real numbers indeed have the property that next to 0 there is no next number, but not every uncountable set has that property.

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Thanks for taking the time to answer my questions

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Hmm, not exactly, because finite sets can also not be put in one-to-one correspondance with the naturals. But you could define uncountable as that the set is finite and you can't put it into a one-to-one correspondence with the naturals.so uncountable means that I cant put the set into a one-to-one correspondence with the natural numbers.

Yes!And are all uncountable sets infinite?

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If I excluded finite sets would that definition work . And I am still not sure what a dense set is.

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Yes, then it would work.If I excluded finite sets would that definition work.

A set D is dense in X if [itex]\overline{D}=X[/itex]. It has nothing to do with countability, but everything with topology and metric spaces.And I am still not sure what a dense set is.

A characterization for metric spaces that I find very helpful:

A set D is dense in X if for every x in X, there exists a sequence (x

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X can be the underlying metric space, or topological space. And D is a subset of X. For example, we could have [itex]X=\mathbb{R}[/itex] and [itex]D=\mathbb{R}\setminus \{0\}[/itex].Im not sure I understand you definition of dense . When you say a set D is dense in X, What is X is it a set,

What do you mean, "match up the elements"? I mean that xand then you say x_n goes to x , Are you saying I can match up these elements?

[tex]\forall \varepsilon >0:~\exists n_0:~\forall n\geq n_0:~d(x,x_n)<\varepsilon[/tex].

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is x an element of D . and when you say [itex] x_n [/itex] converges to x , is this like a limit ?

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No, x does not need to be an element of D.is x an element of D .

Yes, this is like a limit.and when you say [itex] x_n [/itex] converges to x , is this like a limit ?

I should have asked you a long time ago: but what math classes did you already take? And what made you ask this question. Maybe we can give you a better answer depending on that information...

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OK, thanks. But what made you ask this question? Where did you encounter the notion of "dense set"?

And no, it isn't possible for the union of two dense sets to be non-dense...

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Hmm, in that point-of-view, I'm doubting that the author of the book uses the same dense as we use. There are more definitions of "density" out there, so perhaps he's using another one.I was reading a book called infinity and it talked about dense sets.

No, uncountable means that the set is big. And the union of two big sets is an even bigger set. Thus it remains uncountable.Could I have the union of 2 uncountable sets and make it a countable set.

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Is the smallest infinity the set of natural numbers?

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Yes!Is the smallest infinity the set of natural numbers?

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There are as many positive even numbers as there are natural numbers. Indeed, we have a one to one correspondence:

[tex]\mathbb{N}\rightarrow\{\text{even numbers}\}:n\rightarrow 2n[/tex]

And if there is a one-to-one correspondence between two sets, then these sets have equal size. It might seems paradoxical that a proper subset has as many elements as the superset, but that's something we have to live with. It is a situation that arises whenever we deal with infinity.

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