Pasting Lemma/Continuous Mappings on Open Sets

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Theorem

Let $T = \struct {X, \tau}$ and $S = \struct {Y, \sigma}$ be topological spaces.

Let $I$ be an indexing set.

Let $\family {C_i}_{i \mathop \in I}$ be a family of open sets of $T$.

Let $\family {f_i : C_i \to Y}_{i \mathop \in I}$ be a family of continuous mappings.


Let $C = \ds \bigcup_{i \mathop \in I} C_i$.

Let $f = \ds \bigcup_{i \mathop \in I} f_i : C \to Y$ where $\ds \bigcup_{i \mathop \in I} f_i$ is the union of relations.


Let for all $i, j \in I$, $f_i$ and $f_j$ agree on $C_i \cap C_j$.


Then $f$ is a continuous mapping on $C = \ds \bigcup_{i \mathop \in I} C_i$.


Proof

From Union of Family of Mappings which Agree is Mapping:

$f$ is a mapping from $C$ to $Y$.

From Restriction of Union of Mappings which Agree Equals Mapping:

$\forall i \in I : f \restriction_{C_i} = f_i$

Hence:

$\forall i \in I : f \restriction_{C_i}$ is continuous

From Pasting Lemma for Union of Open Sets

$f$ is continuous.

$\blacksquare$


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