5

I have a question about isolated points. Here is my definition.

A point $a \in A \subseteq \mathbb{R}$ is said to be an isolated point of the set $A$ provided there is an open interval $(c,d)$ such that $(c,d) \cap A = \{a\}$.

I need to find the isolated points of $(0,1)$, the set of natural numbers, $\mathbb{N}$, and the set $\{1/n : n \in \mathbb{N}\}$.

Can someone help me understand and apply the definition of a isolated point?

Michael Hardy
  • 1
Carl
  • 93
  • 2
    The idea is to find an interval around the isolated point which does not contain any other element of the set. – Alex Becker May 09 '14 at 20:50
  • As a starting point try to determine whether $(0, 1)$ contains any isolated points according to your definition. Then try to answer whether the set of natural numbers and the set ${1 / n : n \in \mathbb{N}}$ contain any isolated points according to your definition. – Raj May 09 '14 at 20:54
  • I am curious, what metric are you using on the first set? – IAmNoOne May 09 '14 at 21:01

1 Answers1

2

Take $x\in (0,1).$ Then the intersection of any open interval $(c,d)$ (assume $0<c<d<1$) with $(0,1)$ is $(c,d),$ which is different of $x.$ This shows that $x$ is not isolated. So, there is not isolated points in $(0,1).$

Any natural number is isolated. Take an interval of the form $(n-1,n+1)$ and you have $(n-1,n+1)\cap \mathbb{N}=\{n\}.$

Any point of $A=\{1/n/n\in \mathbb{N}\}$ is isolated. Consider the interval $(1/(n+1),1/(n-1))$ and then you have $(1/(n+1),1/(n-1))\cap \mathbb{N}=\{1/n\}.$

mfl
  • 29,399