Proposition 10.3.2.label Let $\seqi{X}$ and $X$ be zero-dimensional spaces, then:

1. (1)

   For any $A \subset X$, $A$ is zero-dimensional.

2. (2)

   $\prod_{i \in I}X_{i}$ is zero-dimensional.

3. (3)

   $\bigsqcup_{i \in I}X_{i}$ is zero-dimensional.