Covering of a universal space and covering of a subset

Question

It seems to me that the definition of covering depends on the set we are referring to. For example, if $X$ is the universal topological space, then a collection $\mathcal A$ of subsets of $X$ is said to be a covering of $X$ if the union of the elements of $\mathcal A$ is ${\bf\underline{equal}}$ to $X$. However, if $Y$ is a subset of $X$, then a covering of $Y$ is a collection $\mathcal B$ of subsets of $X$ union of whose elements ${\bf\underline{contains}}$ $Y$. Is this correct, please? Thank you! I found it is rather confusing in many proofs.

Answer 1

This apparent oddity comes from the fact that every open set is a subset of $X$, which is not true for $Y$. We can combine these definitions by saying "contains" every time, since contains immediately implies equals for a universal space.