Disjoint unions and definition Definition:
Suppose $(X_i)_i$ is an indexed family of non-empty topological spaces.  There is a canonical injection $\sigma_i: X_i \rightarrow \coprod_{i\in I}X_i$ , given by $\sigma_i(x)=(x,i)$. 
The author states that it is a convention to identify $X_i$ with $\sigma_i(X_i)$
Let $X=\coprod_{j\in J}X_j$
The topology is on the disjoint union is defined as
$\tau$ $=$ $\{$ $U\subseteq \ X$ $:$ $U\cap X_j$ (as a subset of X) is open in $X_j$ for each $j\in J$ $\}$
My understanding is as follows:
without identifying the sets, we get that for $U\subseteq X$:
$x\in \sigma_j^{-1}(U) \iff$ $x\in X_j$ and $\sigma_j(x)\in U$ $\iff$ $x\in X_j$ and $(x,j)\in U$ $\iff$ $(x,j)\in \sigma_j(X_j)$ and $(x,j)\in U$ $\iff$ $(x,j)$ $\in \sigma_i(X_j)$ $\cap$ $U$ . 
Identifying the sets means that we treat $\sigma_j^{-1}(U)$ as $\sigma_j(X_j) \cap U$ and vice versa.
So, the topology is really defined to be:
$\tau$ $=$ $\{$ $U\subseteq \ X$ $:$ $\sigma_j^{-1}(U)$ is open in $X_j$ for each $j\in J$ $\}$
(Without the identification)
**Is my understanding correct? 
May I have advice on how to think about these? Why is the author identifying sets? Why not just write them "normally"? 
 A: Yes, you are correct. The topology is the final topology wrt the standard injections $\sigma_i, i \in I$ on the space $\coprod_{i \in I} X_i$ (defined as the standard set disjoint union construction).
It's easy to observe that each separate map $\sigma_{i_0}$ is also open, as $(\sigma_i)^{-1}[\sigma_{i_0}[O]]= O$ for $i=i_0$ and $\emptyset$ if $i \neq i_0$, and when $O \subseteq X_{i_0}$ is open, this set is open for all $i$ for and hence is sum-open in the final topology. So $\sigma_i[X_i]$ is homeomorphic to $X_i$ for all $i$, hence the identification.
And then you can reformulate $O$ being open in $\coprod_{i \in I} X_i$ as 
$$\forall i \in I: O \cap \sigma_i[X_i] \text{ open in } \sigma[X_i]$$
because $$O \cap \sigma_i[X_i] = \sigma_i^{-1}[O]$$
So that $\coprod_{i \in I} X_i$ has the so-called coherent topology wrt its subspaces $\sigma_i[X_i]$ (after we've declared $\sigma_i$ to be a homeomorphism, so making the identification).
So this gives two views on the topology, that come down to the same idea in the end.
