Inclusion Mappings to Topological Sum from Components

Theorem
Let $\left({X, \vartheta_1}\right)$ and $\left({Y, \vartheta_2}\right)$ be topological spaces.

Let $\left({Z, \vartheta_3}\right)$ be the topological sum of $X$ and $Y$ where $\vartheta_3$ is the topology generated by $\vartheta_1$ and $\vartheta_2$.

Then $\vartheta_3$ is the finest topology on $Z$ in which the inclusion mappings from $\left({X, \vartheta_1}\right)$ and $\left({Y, \vartheta_2}\right)$ to $\left({Z, \vartheta_3}\right)$ are continuous.