Definition:Topological Sum
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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