Definition 3.1.5 (Covering).label Let $A\suf X$ be subset of a topological space. A collection $(U_{i})_{i\in I}$ is called a covering of $A$ if $A\suf\cups{}{i\in I}U_{i}$ and $U_{i}\suf X$ for each $i\in I$. The covering is said to be open if all $U_{i}$ are open in $X$.