1. ## 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?

2. ## Re: Density and completeness

Originally Posted by rokman54
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 $\mathbb{R}$.