TITLE: Wedge sum of an empty set of spaces
QUESTION [5 upvotes]: In algebraic topology wedge sum is defined for  a family of indexed, say by a set $\mathcal{I}$, spaces $X_i$ with points $p_i \in X_i$ as:
$$
 \bigvee_{i \in \mathcal{I}} X_i = \frac{\bigsqcup_{i \in \mathcal{I}} X_i}{(i,p_i)\sim(j,p_j)}.
$$
Exactly the same definition will work just for the pointed sets. But what happens then $\mathcal{I} = \emptyset$?
One resolution comes form the fact that wedge sum behaves as coproduct in the category of pointed topological spaces $\mathsf{TOP}^*$, or equivalently pointed sets $\mathsf{SET}^*$, for a reasonable (meaning $\mathcal{I} \neq \emptyset$) families of objects . Then in case of $\mathcal{I} = \emptyset$ the coproduct, if it exists, must be equal to the initial object of the corresponding category. So, in case with $\mathsf{SET}^*$, the zero object, so
$$
 \bigvee_{i \in \emptyset} X_i = \Big(\{p\}, p  \Big),
$$
and the same resolution works for $\mathsf{TOP}^*$.
Is this a standard definition? Is there any reasons not to use it? Do algebraic topologists have any other conventions.
The only thing that worries me here is that together with the original elementary definition the resolution suggests a weird identity
$$
\frac{\emptyset}{\emptyset} = \bigvee_{i \in \emptyset} X_i = \{p\}
$$

REPLY [5 votes]: In any category the coproduct over an empty indexing set is the initial object (assuming existence of things), so you are right that in $\mathsf{Top}_\ast$ we have
$$\bigvee \limits_{i\in\emptyset} (X_i, x_i) = (\{*\},*)$$
Your unintuitive identity doesn’t make sense to me, since $\emptyset$ is uninhabited / empty, hence cannot be regarded as a pointed topological space. But anyway you are right in the assumption that $$\emptyset/\emptyset=*$$ is a common convention, usually seen in the context of comparing reduced- and unreduced (co)homology theories…