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.