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Bachelier
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[tex]\ Is \ \mathbb{N} \ dense \ in \ itself.[/tex]
HallsofIvy said:"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 said:So is there a point [itex]p\in(n,n+1)\forall n\in\mathbb{N}[/itex] such that [itex]p\in\mathbb{N}[/itex]?
micromass said: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.
What kind of 'denseness' do you have in mind here? Some measure theoretic concept?micromass said: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.
The natural numbers, also known as the counting numbers, are the set of positive integers starting from 1 and continuing infinitely. They are represented by the symbol N or N0.
For the natural numbers to be dense in itself means that between any two natural numbers, there exists another natural number. In other words, there are no "gaps" or missing numbers in the set of natural numbers.
The proof for this statement is based on the fact that the natural numbers are an infinite set. We can use a proof by contradiction, assuming that there exists a gap between two natural numbers, and then show that this assumption leads to a contradiction. This proves that there cannot be any gaps between natural numbers, thus proving that they are dense in itself.
Yes, the natural numbers are dense in other number systems such as the integers, rational numbers, and real numbers. However, they are not dense in the set of complex numbers.
The Archimedean property is an example of a property that relies on the natural numbers being dense in itself. It states that for any two positive real numbers, there exists a natural number such that when multiplied by the first number, the result is greater than the second number. This property relies on the fact that there are no gaps between natural numbers.