The wedge of a family of pointed spaces

Let $((X_i,x_i){)}_{i\in I}$ be a family of pointed topological spaces. On the tagged disjoint union ${\coprod }_{i\in I}{X}_i$ (The disjoint union (coproduct) ${⨆}_i{X}_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is), define

$(x,i)\sim (y,j)⟺(x,i)=(y,j)\text{ or }(x=x_i\text{ and }y=x_j)\mathrm{.}$

This relation is reflexive and symmetric. For transitivity, the only nontrivial case has two related pairs that are not equal; then every point appearing is its summand's basepoint, so the first and third points are related as well. Thus $\sim$ is an equivalence relation.

For nonempty $I$, the wedge is the quotient space

${\bigvee }_{i\in I}(X_i,x_i):=\left({\coprod }_{i\in I}{X}_i\right)\text{\hspace{0.17em}⁣}/\sim$

with the quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection). The common equivalence class of the tagged basepoints is the wedge point, which makes the quotient pointed. The relation identifies no other points.

The wedge of the empty family is defined to be a one-point space, pointed at its sole element. For a pair one writes $(X,x_0)\vee (Y,y_0)$, and for $r\in \mathbb{N}$ one writes ${\bigvee }_{j<r}(X_j,x_j)$. In particular, the empty finite wedge is a point rather than an empty space.