Rudin Principles of Mathematical Analysis
2.18 Definition Let X be a metric space. All points and sets mentioned below are understood to be elements and subsets of X.
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(j) E is dense in X if every point of X is a limit point of E, or a point of E (or both).
Rudin does not offer any examples of this in the entire chapter. I am trying to get an example in order to understand this.
If we let X:=${\mathbb R}$ , and E be a subset, say the interval I:=(a,b), with a=0 and b=1 for example. Then every point of E=I is a limit point?
Does this satisfy the definition? I could use some guidance.
Best Answer
A working idea of "denseness" in your context -- with the usual metric in $\mathbb{R}$:
$E$ is dense in $X$ if $E$ plus it's limit points =$X$.
I believe you are confusing the issue of which set the limit points come from.
Every point in $X$ needs to be in $E$ or a limit point of a sequence from $E$.