Pasting Lemma/Pair of Continuous Mappings on Open Sets/Proof 1
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Theorem
Let $X$ and $Y$ be topological spaces.
Let $A$ and $B$ be open in $X$.
Let $f: A \to Y$ and $g: B \to Y$ be continuous mappings that agree on $A \cap B$.
Let $f \cup g$ be the union of the mappings $f$ and $g$:
- $\forall x \in A \cup B: \map {f \cup g} x = \begin {cases} \map f x & : x \in A \\ \map g x & : x \in B \end {cases}$
Then the mapping $f \cup g : A \cup B \to Y$ is continuous.
Proof
Follows directly from Pasting Lemma for Continuous Mappings on Open Sets.
$\blacksquare$