Inclusion Mappings to Topological Sum from Components
Theorem
Let $\left({X, \vartheta_1}\right)$ and $\left({Y, \vartheta_2}\right)$ be topological spaces.
Let $\left({Z, \vartheta_3}\right)$ be the topological sum of $X$ and $Y$ where $\vartheta_3$ is the topology generated by $\vartheta_1$ and $\vartheta_2$.
Then $\vartheta_3$ is the finest topology on $Z$ in which the inclusion mappings from $\left({X, \vartheta_1}\right)$ and $\left({Y, \vartheta_2}\right)$ to $\left({Z, \vartheta_3}\right)$ are continuous.
Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions
