Definition 0: Let $Y\subseteq X$. Let $\lbrace A_i \rbrace_{i\in I}=\lbrace B_\alpha\subseteq X\mid\alpha\in J \rbrace$. Thus, $\lbrace A_i \rbrace_{i\in I}$ covers $Y$ iff $Y\subseteq \bigcup_{\alpha \in I}A_\alpha$.
Definition 1: Let $(X, T)$ be a topological space. Now, let $Y\subseteq X$. Let $T_1=\lbrace Y\cap U\mid U\in T \rbrace$. Thus, $T_1$ on $Y$ is induced by the topology $T$ on $X$ which yields a topological space $(Y, T_1)$.
Definition 2: Now, let $X$ be topological space. Let $Y\subseteq X$. A covering $\lbrace A_\alpha \rbrace_{\alpha \in I}$ of $Y$ is said to be an open covering of $Y$ iff $\forall \alpha \in I$, "$A_\alpha$ is an open subset of $X$".
Question: Saying "$A_\alpha$ is an open subset of $X$" is confusing me, because there are multiple possibilities where the open subsets of $X$ could come from. Instead of saying "$A_\alpha$ is an open subset of $X$" which option below $\textbf{(A.)}$, $\textbf{(B.)}$, or $\textbf{(C.)}$ is it?
$\textbf{(A.)}$ "$A_\alpha$ is an element in the topology $T_1$ on $Y$ where $T_1$ on $Y$ is induced
$\text{ }$ by the topology $T$ of $X$ (aka $A_\alpha\in T_1$)"
$\textbf{(B.)}$ "$A_\alpha$ is an element in the topology of $X$ (aka $A_\alpha\in T$)"
$\textbf{(C.)}$ "$A_\alpha$ is an either of these topologies (it doesn't matter)"
Best Answer
Well, it can be either A or B but if we go by the second definition you provided, it is B:
Now in order for it to be A, then any question must specify where those open sets lie. Since your definition says that they are open in $X$ then so it is. If it doesn't specify, but you're given a topology only on $X$ it should be safe to assume that $A_{\alpha}$ are open in $X$. Else, it should specify that it's an open cover of $Y$ in the $T_1$ topology.