Open Cover and Subcover of a Subset of a Topological Space

Let $(X,\mathcal{T})$ be a topological space, let $A\subseteq X$, let $I$ be a set, and let $(U_i)_{i\in I}$ be a family of subsets of $X$.

We say that $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ if the following two conditions hold.

1. For every $i\in I$, one has $U_i\in\mathcal{T}$.
2. One has

$$A\subseteq \bigcup_{i\in I} U_i.$$

If $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ and if $J\subseteq I$, then the subfamily $(U_j)_{j\in J}$ is called a subcover of $A$ if

$$A\subseteq \bigcup_{j\in J} U_j.$$