Definition:Topological Sum

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

The topological sum $\left({Z, \vartheta_3}\right)$ of $X$ and $Y$ is defined as:

where $X \sqcup Y$ denotes the disjoint union of $X$ and $Y$;
 * $Z = X \sqcup Y$


 * $\vartheta_3$ is the topology generated by $\vartheta_1$ and $\vartheta_2$.

Also see

 * Inclusion Mappings to Topological Sum from Components, in which it is demonstrated that the topology $\vartheta_3$ has the property that it is the finest topology on $Z$ such that the inclusion mappings from $\left({X, \vartheta_1}\right)$ and $\left({Y, \vartheta_2}\right)$ to $\left({Z, \vartheta_3}\right)$ are continuous.