Prove that $U=x in Xmid d(x,A)<d(x,B)$ is open when $A$ and $B$ are disjoint
Clash Royale CLAN TAG #URR8PPP 4 $begingroup$ In a metric space $(X,d)$ , I have the set $$U=x in Xmid d(x,A)<d(x,B)$$ where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$ . I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(x,A)=d(x,C)$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you! metric-spaces share | cite | improve this question edited Mar 14 at 19:47 Asaf Karagila ♦ 308k 33 441 775 asked Mar 14 at 13:34 Sean Thrasher Sean Thrasher 44 4 $endgroup$ add a comment | 4 $begingroup$ In a metric space $(X,d)$ , I have the set $$U=x in Xmid d(x,A)<d(x,B)$$ where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$ . I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(...