Question

the union of two open set is an open set

Answer 1

union of two open set is an open set

Answer 2

Let A1,A2A1,A2 open sets, we need to prove A1∪A2A1∪A2 is open. As A1,A2A1,A2 are open set, then for all a1,a2∈A1,A2a1,a2∈A1,A2 respectively we have r1,r2>0r1,r2>0 such that B(a1,r1)⊂A1B(a1,r1)⊂A1 and B(a2,r2)⊂A2B(a2,r2)⊂A2 Let r=12min{r1,r2}r=12min{r1,r2} and x∈A1∪A2x∈A1∪A2, then x∈A1x∈A1 or x∈A2x∈A2 This implies: B(x,r)⊂A1∪A2B(x,r)⊂A1∪A2 In conclusion, A1∪A2A1∪A2 is open set.