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... $\displaystyle \overline{S} = [0, 1]$?

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.

Re: Quick Cluster Points Question

Quote:

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]

Re: Quick Cluster Points Question

Quote:

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.

Re: Quick Cluster Points Question

Quote:

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.