Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains an open ball as well.
Thank you!
Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains an open ball as well.
Thank you!
Hint: Baire category theorem. If no $A_k$ contains an open ball, then all $A_k$'s are nowhere dense, hence $\bigcup_{k} A_k$ ...