subspace of wedge product without point as coproduct

Let \(X_i\) be based spaces and \(\bigvee X_i\) the wedge product. Then the subspace \(\bigvee X_i \setminus *\) is homeomorphic to the topological sum

\begin{align*} \bigvee X_i \setminus * \cong \coprod (X_i \setminus *) \end{align*}

Consider the map

\begin{align*} \iota_i: X_i \setminus * \rightarrow \bigvee X_i \setminus * x_i \mapsto& x_i \end{align*}

is is continuous and open by the definition of the quotient topology.

Thus by glueing lemma for open maps on an open covering and glueing lemma for an open covering it follows, that

\begin{align*} \coprod \iota_i: X_i \setminus * \rightarrow \bigvee X_i \setminus * \end{align*}

is both open and continuous. Furthermore, it is clearly bijective thus an isomorphism

existence of an abstract proof ?