Notation for the disjoint union of open subspaces Given a family $(X_i)_{i\in I}$ of pairwise disjoint open subspaces $X_i\subset X$ of a topological space $X$, their union is canonically homeomorphic to the topological disjoint union $\bigsqcup_{i\in I} X_i$, which, as a set, is defined as
$$ \bigsqcup_{i\in I} X_i := \bigcup_{i\in I} X_i \times \{i\}. $$
Strictly speaking, this space is not a subset of $X$. I am wondering if there is a notation to emphasize that the union $\bigcup_{i\in I} X_i \subset X$ has the disjoint union topology, as a subspace of $X$. Is it customary to write $\bigsqcup_{i\in I} X_i$ also for this subspace?

Yes, it is customary to write $\coprod X_i$ for a disjoint union of subspaces/subsets in general. And $\coprod X_i$ is also the usual notation for the abstract disjoint union of the $X_i$ (which is the coproduct in the category of spaces - this notation is also used for (categorical) coproducts in general).  This usually does not lead to confusion, and in any case, whenever you feel that it can be confusing, you should say explicitely what you are speaking about. A good way to avoid confusion is also to prefer using only $\bigcup X_i$ for the subspace if you want to distinguish it from the abstract disjoint union.
