# Metric Space , Close and open

#### 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

• steve2009

#### Jose27

MHF Hall of Honor
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 $$\displaystyle \mathbb{Q}$$?
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 $$\displaystyle \mathbb{Q}$$?