Quick Cluster Points Question

Printable View

July 4th 2011, 09:50 AM
jameselmore91

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, $S = Q \bigcap [0, 1]$ , then no point in S is an interior point of S and $\overline{S} = [0, 1]$ "

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 $\overline{S}$ . Or are they saying that no point in S is an interior point of S... AND... $\overline{S} = [0, 1]$ ?
July 4th 2011, 09:55 AM
jameselmore91

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.
July 4th 2011, 10:03 AM
Plato

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]
July 4th 2011, 10:09 AM
Plato

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, $S = Q \cap [0, 1]$ , then no point in S is an interior point of S and $\overline{S} = [0, 1]$ "
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 $\overline{S}$

Any open set that contains a point of $S$ also contains an irrational number. Therefore every point of $S$ is a boundary point if $S$ and not an interior point.
July 4th 2011, 10:54 AM
HallsofIvy

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, $S = Q \bigcap [0, 1]$ , then no point in S is an interior point of S and $\overline{S} = [0, 1]$ "

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 $\overline{S}$ . Or are they saying that no point in S is an interior point of S... AND... $\overline{S} = [0, 1]$ ?

It is the second.