# Definition:Topological Sum

Jump to navigation
Jump to search

## Definition

Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces.

The **topological sum** $\struct {Z, \tau_3}$ 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 $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ are continuous.

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

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions