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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 $C=xin Xmid d(x,A)=d(x,C)$ but

Clash Royale CLAN TAG #URR8PPP 33 $begingroup$ org.apache.kafka.clients.consumer.ConsumerRecords@1f6d27cc was empty ScalaTestFailureLocation: SorterTest at (SorterTest.scala:75107) org.scalatest.exceptions.TestFailedException: org.apache.kafka.clients.consumer.ConsumerRecords@1f6d27cc was empty at org.scalatest.Assertions.newAssertionFailedException(Assertions.scala:6597) at org.scalatest.Assertions.newAssertionFailedException $(Assertions.scala:70102) at org.scalatest.FunSuite.newAssertionFailedException(FunSuite.scala:75107) at org.scalatest.Assertions$ AssertionsHelper.macroAssert(Assertions.scala:6597) at SorterTest. $anonfun$ new $4(SorterTest.scala:72104) at SorterTest.$ anonfun $new$ 4 $adapted(SorterTest.scala:6597) at scala.collection.immutable.List.foreach(List.scala:84116) at SorterTest.$ anonfun $new$ 3(SorterTest.scala:69101) at scala.runtime.java8.JFunction0 $mcV$ sp.apply(JFunction0 $mcV$ sp.java:83115) at org.scalatest.OutcomeOf.outcomeO