# Thread: Quick Cluster Points Question

1. ## Quick Cluster Points Question

I'm beginning to study Analysis but I'm having the wording of this particular paragraph is confusing to me:

"The definition of boundary point (when this is not an isolated point) implies that any ball about such a point actually contains infinitely many points from S. Thus, a cluster point can be either a boundary point or an interior point of a set S but never an exterior point of S. If, for instance S is the set of rational numbers contained in the unit interval, $\displaystyle S = Q \bigcap [0, 1]$, then no point in S is an interior point of S and $\displaystyle \overline{S} = [0, 1]$"

It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of $\displaystyle \overline{S}$. Or are they saying that no point in S is an interior point of S... AND... $\displaystyle \overline{S} = [0, 1]$?

2. ## Re: Quick Cluster Points Question

I'm getting LaTeX errors on most of what I put in there... can others see it? I don't know what the deal is.

3. ## Re: Quick Cluster Points Question

Originally Posted by jameselmore91
I'm getting LaTeX errors on most of what I put in there... can others see it? I don't know what the deal is.
Use [TEX]...[/TEX] not [Math]...[/Math]

4. ## Re: Quick Cluster Points Question

Originally Posted by jameselmore91
. If, for instance S is the set of rational numbers contained in the unit interval, $\displaystyle S = Q \cap [0, 1]$, then no point in S is an interior point of S and $\displaystyle \overline{S} = [0, 1]$"
It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of $\displaystyle \overline{S}$
Any open set that contains a point of $\displaystyle S$ also contains an irrational number. Therefore every point of $\displaystyle S$ is a boundary point if $\displaystyle S$ and not an interior point.

5. ## Re: Quick Cluster Points Question

Originally Posted by jameselmore91
I'm beginning to study Analysis but I'm having the wording of this particular paragraph is confusing to me:

"The definition of boundary point (when this is not an isolated point) implies that any ball about such a point actually contains infinitely many points from S. Thus, a cluster point can be either a boundary point or an interior point of a set S but never an exterior point of S. If, for instance S is the set of rational numbers contained in the unit interval, $\displaystyle S = Q \bigcap [0, 1]$, then no point in S is an interior point of S and $\displaystyle \overline{S} = [0, 1]$"

It's the last sentence that loses me. Are they saying that no point in S is an interior point of S and an interior point of $\displaystyle \overline{S}$. Or are they saying that no point in S is an interior point of S... AND... $\displaystyle \overline{S} = [0, 1]$?
It is the second.