# Definition:Topological Sum

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## Definition

Let $\left({X, \tau_1}\right)$ and $\left({Y, \tau_2}\right)$ be topological spaces.

The **topological sum** $\left({Z, \tau_3}\right)$ of $X$ and $Y$ is defined as:

- $Z = X \sqcup Y$

where:

- $X \sqcup Y$ denotes the disjoint union of $X$ and $Y$
- $\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$.

## Also see

- Inclusion Mappings to Topological Sum from Components, in which it is demonstrated that the topology $\tau_3$ has the property that it is the finest topology on $Z$ such that the inclusion mappings from $\left({X, \tau_1}\right)$ and $\left({Y, \tau_2}\right)$ to $\left({Z, \tau_3}\right)$ are continuous.

- Results about
**topological sum**can be found here.

## Sources

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