Inclusion Mappings to Topological Sum from Components

Theorem
Let $\left({X, \tau_1}\right)$ and $\left({Y, \tau_2}\right)$ be topological spaces.

Let $\left({Z, \tau_3}\right)$ be the topological sum of $X$ and $Y$ where $\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$.

Then $\tau_3$ is the finest topology on $Z$ in which the inclusion mappings from $\left({X, \tau_1}\right)$ and $\left({Y, \tau_2}\right)$ to $\left({Z, \tau_3}\right)$ are continuous.