"Topological disjoint union" vs "disjoint union" I want to understand the definition of topological disjoint union. Let $X$ be a topological space such that we can write $X=\bigsqcup X_i$. We say that $X$ is the "topological disjoint union" of the $X_i$ if each $X_i$ is an open subset of $X$, but if the $X_i$ are not necesserily open subsets of $X$ we say that $X$ is a the "disjoint union" of the $X_i$ only as sets (without the topological adjective) which means that $X =\cup X_i$ is the union of the $X_i$ such that all the $X_i$ are pairwise disjoint as sets. Is my understanding correct? thank you for your help!
 A: Yes, you are correct. But sometimes the adjective “topological” is implied as you are working in the category of topological spaces. One can make the distinction also by saying “disjoint union of subspaces” vs. “disjoint union of subsets”. It really depends on the context how careful one has to be to distinguish the two situations.
I would also like to add that there two versions of the sum / disjoint union. One of them is a construction – you start with any family of sets/spaces, and then you construct their sum. For this you often have to replace the original spaces by isomorphic copies that are disjoint. The other one is a property – you already have the whole space and a family of its subspaces, and you ask whether the whole space is the sum of those subspaces. Of course the two variants are closely connected – the latter propery holds if and only if the given space is canonically isomorphic to the constructed one. Also note that this phenomenon is not special to topological spaces and sets, it happens for example with the direct sum of modules / vector spaces.
A: A topological space is a pair $(X,T)$ where $T$ is a topology on the set $X,$ but it is common to call it "the space $X$" when a particular $T$ is assumed.
If $(X_i,T_i)$ is a topological space for each $i\in I$ then the topological sum (topological disjoint union) is $(X, T)$ where $X=\cup_{i\in I} X_i\times \{i\}$ and $T$ is the topology on $X$ generated by the base (basis) $\cup_{i\in I}\{t\times \{i\}: t\in T_i\}.$
