What is the "empty cover"?

I am studying metric spaces and I have the following definitions for an open cover of a metric space/subset of a metric space:

An open cover of a metric space ($X,d$) is a family of open sets $(U_\alpha)_{\alpha \in I}$ such that $\bigcup_{\alpha \in I} U_{\alpha} = X$

An open cover of a subset $S$ of a metric space $X$ is a family of open sets $(U_\alpha)_{\alpha \in I}$ such that $\bigcup_{\alpha \in I} U_{\alpha} \supseteq X$

I read online that the "empty cover" is an open cover of the empty set. I assume the empty cover is a family of sets where $I = \emptyset$. I guess its vacuously true that every set in this family is open, but how can it be true that $\bigcup_{\alpha \in I} U_{\alpha} \supseteq \emptyset$? How can you define the union of no sets?

The statement $\emptyset \subseteq X$ is true for all sets $X$.

This is because, by definition, the statement $A\subseteq B$ is equivalent to the statement $$\forall a: a\in A\implies a\in B.$$

If $A=\emptyset$ and $B=X$, then $A\subseteq B$ is true if and only if the following is true:

$$\forall x: x\in\emptyset\implies x\in X.$$

This statement is obvioulsy true, because "$x\in\emptyset$" is always false, and if $p$ is false, then $p\implies q$ is always true.