Continuous Mapping of Separation
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Theorem
Let $T$ and $T'$ be topological spaces.
Let $A \mid B$ be a separation of $T$.
Let $f: T \to T'$ be a mapping such that the restrictions $f \restriction_A$ and $f \restriction_B$ are both continuous.
Then $f$ is continuous on the whole of $T$.
Proof
Follows directly from Pasting Lemma for Union of Open Sets.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Proposition $6.2.5$