Metric Space , Close and open

Sep 2009
21
2
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
 
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Jose27

MHF Hall of Honor
Apr 2009
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México
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 \(\displaystyle \mathbb{Q}\) 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 2009
21
2
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.

thank you for your reply