# Metric Space , Close and open

• Sep 28th 2009, 06:00 PM
donsmith
Metric Space , Close and open
We have a, b ∈ R \ Q, a < b.
A = {x ∈ Q: a < x < b}

Show that A is clopen(open and close) in Q

i have being trying to solve this problem in the last 2hours

i need some hints or examples

• Sep 28th 2009, 06:22 PM
Jose27
Quote:

Originally Posted by donsmith
We have a, b ∈ R \ Q, a < b.
A = {x ∈ Q: a < x < b}

Show that A is clopen(open and close) in Q

i have being trying to solve this problem in the last 2hours

i need some hints or examples

Assuming you give $\displaystyle \mathbb{Q}$ the topology inherited from $\displaystyle \mathbb{R}$ then open sets in $\displaystyle \mathbb{Q}$ are those of the form $\displaystyle A \cap \mathbb{Q}$ where $\displaystyle A$ is open in $\displaystyle \mathbb{R}$. Is $\displaystyle A_1=\{ x \in \mathbb{R} : a<x<b \}$ open? $\displaystyle A_1$ fails to be closed in $\displaystyle \mathbb{R}$ because it lacks $\displaystyle a$ and $\displaystyle b$, but is that a problem in $\displaystyle \mathbb{Q}$?
• Sep 28th 2009, 07:18 PM
donsmith
Quote:

Originally Posted by Jose27
Assuming you give $\displaystyle \mathbb{Q}$ the topology inherited from $\displaystyle \mathbb{R}$ then open sets in $\displaystyle \mathbb{Q}$ are those of the form $\displaystyle A \cap \mathbb{Q}$ where $\displaystyle A$ is open in $\displaystyle \mathbb{R}$. Is $\displaystyle A_1=\{ x \in \mathbb{R} : a<x<b \}$ open? $\displaystyle A_1$ fails to be closed in $\displaystyle \mathbb{R}$ because it lacks $\displaystyle a$ and $\displaystyle b$, but is that a problem in $\displaystyle \mathbb{Q}$?

is a problem in Q

I don't know how to start my proof.