Definition 3.1.4 (Covering).label Let $X$ be a set and $A\suf X$. A collection $\curl{U_i}_{i\in I}$ is called a covering of $A$ if $A\suf\cups{}{i\in I}U_{i}$ and $U_{i}\suf X$ for each $i\in I$. A covering $\curl{U_i}_{i\in I}$ is said to be open (resp. closed) if $X$ is a topological space and all $U_{i}$ are open (resp. closed) in $X$. A covering $\curl{U_i}_{i\in I}$ is said to be (pairwise)disjoint if for each $i,j\in I$, $U_{i}\cap U_{j}=\begin{cases}\emp&i\neq j\\ U_{i}&i=j\end{cases}$. A subcover of a covering $\curl{U_i}_{i\in I}$ is a covering $\curl{V_j}_{j\in J}$ such that $J\suf I$ and $V_{j}=U_{j}$ for each $j\in J$ and $\curl{V_j}_{j\in J}$ is said to be finite if $J$ is finite.