Does the definition of compactness require the open cover to consist of subsets? Sorry if the title is a bit unclear, but I'm stuck on the definition of compactness for metric spaces using open covers. So our professor wrote (word for word) that in a metric space $(X,d)$, an open cover is a family $\{U_i\}_{i\in I}$ with
$U_i\subset X$ open, 
$\bigcup_{i\in I}U_i=X$
and that $X$ is compact if from every open cover $\{U_i\}_{i\in I}$ finitely many $U_{i_1},...,U_{i_r}$ can be chosen such that $U_{i_1}\cup U_{i_2} \cup ... \cup U_{i_r}=X$.
Is this definition correct? Because in one of our exercises, they ask us to find an open cover (that doesn't have a finite subcover) for $X=[0,1]\cap \Bbb Q$ with the Euclidian distance, but I can't for the life of me think of any set that is a subset of $X$, contains $0$ or $1$, and on top of that is an open set - those three things seem mutually exclusive to me. Was our prof mistaken in writing that $U_i$ is a subset of $X$ in the definition for open covers?
Any help would be greatly appreciated!
 A: Open means relatively open. A subset of $U$ of $X = [0,1] \cap \mathbf Q$ is open if there is an open set $O \subset \mathbf R$ satisfying $U = O \cap X$.
A: A subset $K$ of a metric space $X$ is compact if and only if for every any collection of open sets $U_i\subseteq X,i\in I$ for which $K\subseteq \bigcup_{i\in I} U_i$, there is a finite set $F\subseteq I$ for which $K\subseteq \bigcup_{i\in F}U_i$. Note that $U_i$ need only be subsets of $X$.
However, $K$ being compact as a subset of $X$ is equivalent to $(K,d|_K)$ being a compact space. Here, $d|_K$ is the metric on $X$ with its domain restricted to $K\times K$. It can be shown that $V\subseteq K$ is open in $(K,d|_K)$ if and only if $V=U\cap K$ for some open $U\subseteq X$. 
For your problem, for example, letting $K=[0,1]\cap \mathbb Q$, then $(a,b)\cap [0,1]\cap \mathbb Q$ would be an open set for any $a,b\in \mathbb R$. Even though such sets are not open in $\mathbb R$, they are open when viewed as subsets of $K$ with the metric inherited from $\mathbb R$.
A: e.g. $[0,\frac12)\cap \mathbb{Q}$ is an open set of $X=[0,1]\cap \mathbb{Q}$ in its subspace topology.
