Metric Space , Close and open

**donsmith** · September 28th 2009, 06:00 PM

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

thank you in advance

**Jose27** · September 28th 2009, 06:22 PM

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

thank you in advance

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

**donsmith** · September 28th 2009, 07:18 PM

Originally Posted by Jose27

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

is a problem in Q

I don't know how to start my proof.

thank you for your reply

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