Product Space is Product in Category of Topological Spaces

Theorem
Let $\mathbf {Top}$ be the category of topological spaces.

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\struct {\XX, \tau}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.

Then $\struct {\XX, \tau}$ is the product of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ in $\mathbf{Top}$.