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:

• $Z = X \sqcup Y$

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

• $\vartheta_3$ 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.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions