# Property of a covering

My problem:

Suppose $$(X,d)$$ is a metric space and $$A=\bigcup_{i \in I}B(x_i,r_i) \subset X$$ is totally bounded where $$B(x,r)=\{y \in X: d(x,y).

I know that if $$A$$ is totally bounded, $$A \subset \bigcup_{j \in \mathbb{N}}B(y_j,3q_j)$$ such that the balls $$B(y_j,q_j)$$ are mutually disjointed.

But in my notes I wrote that if $$A=\bigcup_{i \in I}B(x_i,r_i)$$ where $$I$$ is arbitrary, then we can conclude that $$A=\bigcup_{i \in I}B(x_i,r_i) \subset \bigcup_{n \in \mathbb{N}} B(x_{i_n},5r_{i_n})$$ such that $$B(x_{i_n},r_{i_n})$$ are mutually disjointed.

In other words we can use the same balls of $$A$$ to obtain the covering property.

• check Vitali's Theorem in the book Measure Theory and Fine Properties of Functions, Evans-Gariepy – alphaomega Jul 16 at 12:27