On the several definitions of disjoint union I am confused over the term disjoint union.
I am reading Lee's Introduction to Topological Manifolds, 2011. In his book, he defined disjoint union twice:

*

*The term disjoint union is used when we take union over a family of pairwise disjoint sets.


*The term abstract disjoint union is used when we need to take the union of an indexed family containing not necessarily pairwise disjoint sets but want to keep them disjoint. The elements of the union are ordered pairs of elements and indices of the set.
By these definitions, I would think that the following statement
'The set $ S $ is the disjoint union of $ P $'
implies that $ P $ is a partition of the set $ S $ because if $ P $ is not a partition, the word disjoint could not be used.
But in Wikipedia, it seems that the term disjoint union is used for the second definition by default. If this is the case, the disjoint union of some sets would never guarantee that the sets are pairwise disjoint. Meanwhile, another note on disjoint union exactly points out what I am confused with.
What is the more standard definition of the term? Is it just determined by the context?
 A: Definition 2 is the better technical definition because in many contexts, it shouldn't matter whether the sets are actually disjoint.
For example, say I have two topological spaces $X$ and $Y$ and I want to take their disjoint union $X \coprod Y$. Topology is roughly the study of the properties of topological spaces which are invariant under homeomorphism, so we don't actually care about the specific underlying sets of $X$ and $Y$; instead, we care about the structure.
So it should be irrelevant whether $X$ and $Y$ are actually disjoint when we're doing any kind of topological construction on them. In particular, given two spaces $X$ and $Y$, it's always possible to come up with spaces $X'$ and $Y'$ and homeomorphisms $X \cong X'$, $Y \cong Y'$ such that the underlying sets of $X'$ and $Y'$ are disjoint.
In other contexts, it does potentially matter whether two sets are disjoint. For example, suppose we're given subspaces $X, Y \subseteq Z$. In this case, discussing whether $X$ and $Y$ are disjoint is a legitimate topological question. In this case, if $X$ and $Y$ are disjoint, their disjoint union will be the space $X \cup Y \subseteq Z$.
