Inclusion Mappings to Topological Sum from Components

Theorem
Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces.

Let $\struct {Z, \tau_3}$ 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 $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ are continuous.