The disjoint union (coproduct) $\bigsqcup_i X_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is

The underlying set. Let $I$ be a set and let $X_i$ be a set for each $i \in I$. The disjoint union is

$\bigsqcup_{i \in I} X_i \;:=\; \bigcup_{i \in I} \big(X_i \times \{i\}\big) ,$

whose elements are the pairs $(x, i)$ with $i \in I$ and $x \in X_i$. For $j \in I$ the $j$-th canonical injection is

$\kappa_j : X_j \to \bigsqcup_{i \in I} X_i, \qquad \kappa_j(x) := (x, j).$

The construction is what makes the word "disjoint" honest. Each $\kappa_j$ is injective (Injection, surjection, bijection), since $(x,j) = (x',j)$ forces $x = x'$; the images $\kappa_j[X_j] = X_j \times \{j\}$ are pairwise disjoint, since the second coordinate determines $j$; and their union is the whole set. So no assumption that the $X_i$ are disjoint as sets is needed, and none is made: the tag $i$ separates the copies even when $X_i = X_{i'}$ for $i \ne i'$.

The trace of a subset. For $U \subseteq \bigsqcup_i X_i$ and $j \in I$ write

$U_j \;:=\; \kappa_j^{-1}[U] \;=\; \{\, x \in X_j : (x,j) \in U \,\} \subseteq X_j ,$

the trace of $U$ on the $j$-th summand. A subset is determined by its family of traces, since $U = \bigcup_i \kappa_i[U_i]$.

The topology. Now let each $X_i$ carry a topology $\mathcal{T}_i$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The disjoint union topology (also coproduct topology, or topological sum) on $\bigsqcup_i X_i$ is the final topology of the family $(\kappa_i)_{i \in I}$ (The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology), that is

$\mathcal{T}^{\sqcup} \;:=\; \Big\{\, U \subseteq \bigsqcup_i X_i \;:\; U_i \in \mathcal{T}_i \text{ for every } i \in I \,\Big\} :$

a set is open exactly when each of its traces is open. That this is a topology is discharged in The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology, where the final topology of any family is verified to satisfy (T1), (T2) and (T3); nothing further is needed here.

Closed sets, dually. $F \subseteq \bigsqcup_i X_i$ is closed exactly when every trace $F_i$ is closed in $X_i$. Indeed the trace operation commutes with complementation, $\kappa_i^{-1}[\,\bigsqcup_j X_j \setminus F\,] = X_i \setminus F_i$, so $F$ is closed if and only if the complement is open if and only if every $X_i \setminus F_i$ is open.

Each summand sits inside as a clopen subspace. The set $\kappa_j[X_j] = X_j \times \{j\}$ has traces $X_j$ at $j$ and $\varnothing$ elsewhere, both open and both closed, so it is clopen in the union. Its subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace) is carried across by $\kappa_j$ from $\mathcal{T}_j$, and $\kappa_j$ is an embedding (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); both statements are proved in the next item rather than assumed here.

Degenerate cases. For $I = \varnothing$ the disjoint union is the empty set with its only topology. For $I$ a one-element set the map $\kappa$ is a bijection carrying $\mathcal{T}$ to $\mathcal{T}^{\sqcup}$, so the construction returns the one summand up to homeomorphism and changes nothing.

• Why the tag is part of the element. Writing $\bigsqcup_i X_i$ as the plain union $\bigcup_i X_i$ would collapse points that happen to be shared between two summands, and the two canonical injections would then fail to be injective. Building the tag into the element makes the injectivity, the disjointness and the description "a set is open when each trace is open" true by construction rather than by hypothesis.

• This is the exact dual of the product. The product is an initial topology of maps out of it, the coproduct a final topology of maps into it; the product has the characteristic property for maps into it, the coproduct the characteristic property for maps out of it. Both are instances of The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology and the next item reads the corresponding half of Characteristic properties: a map into a space with the initial topology is continuous iff every composite with the defining family is, a map out of a space with the final topology is continuous iff every composite with the defining family is, and the two topologies are respectively the coarsest and the finest making that family continuous.

• Nothing here is finite. The index set is arbitrary and no choice principle is involved: the injections are given by an explicit formula and the topology is described by a condition on all traces at once, with no selection anywhere.