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Math Help - Density and completeness

  1. #1
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    Density and completeness

    Hey guys,

    I've been seeking quite some help from this forum.

    It has been really helpful in my studies.

    I was wondering what the difference between density and completeness is.

    A dense subset S of R is defined  S \cap I not equal \phi \forall I which is a open interval in R.

    Which basically means that you can draw any interval in R no matter how small, it will intersect with S.

    There will always be an element of S in the real number interval line.

    Completeness of R means that there are no gaps in the real number interval line.

    Basically isn't being complete the same as being dense?
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  2. #2
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    Re: Density and completeness

    Quote Originally Posted by rokman54 View Post
    Hey guys,

    I've been seeking quite some help from this forum.

    It has been really helpful in my studies.

    I was wondering what the difference between density and completeness is.

    A dense subset S of R is defined  S \cap I not equal \phi \forall I which is a open interval in R.

    Which basically means that you can draw any interval in R no matter how small, it will intersect with S.

    There will always be an element of S in the real number interval line.

    Completeness of R means that there are no gaps in the real number interval line.

    Basically isn't being complete the same as being dense?
    Density is a property of the subset S and completeness (as you wrote it) is a property of the ambient space \mathbb{R}. Furthermore, every topological space is dense in itself, even the incomplete ones. On the other hand, the set \{1\} is not dense in \mathbb{R}.
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