Pasting Lemma/Finite Union of Closed Sets

Theorem

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

Let $I$ be a finite indexing set.

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

Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i \in I$.

Then $f$ is continuous on $C = \ds \bigcup_{i \mathop \in I} C_i$, that is, $f \restriction_C$ is continuous.

Corollary 1

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

Let $I$ be a finite indexing set.

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

Let $\family {f_i : C_i \to Y}_{i \mathop \in I}$ be a finite 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 $C = \ds \bigcup_{i \mathop \in I} C_i$.

Corollary 2

Let $X$ and $Y$ be topological spaces.

Let $A$ and $B$ be closed 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

Let $V \subset S$ be a closed set.

By Continuity Defined from Closed Sets, we have that $U_i = \paren {f \restriction_{C_i} }^{-1} \sqbrk V$ is also closed.

From the definition of a restriction, we have that $U_i = C_i \cap f^{-1} \sqbrk V$.

Therefore, we can compute:

$\ds \paren {f \restriction_{C_i} }^{-1} \sqbrk V$

$=$

$\ds C \cap f^{-1} \sqbrk V$

Definition of Restriction of Mapping

$\ds $

$=$

$\ds \paren {\bigcup_{i \mathop \in I} C_i} \cap f^{-1} \sqbrk V$

Definition of $C$

$\ds $

$=$

$\ds \bigcup_{i \mathop \in I} \paren {C_i \cap f^{-1} \sqbrk V}$

Intersection Distributes over Union

$\ds $

$=$

$\ds \bigcup_{i \mathop \in I} U_i$

Definition of $U_i$

That is, $U = \paren {f \restriction_{C_i} }^{-1} \sqbrk V$ is the union of finitely many closed sets.

Therefore, $U$ is itself closed by definition of a topology.

It follows by Continuity Defined from Closed Sets that $f \restriction_C$ is also continuous.

$\blacksquare$

Also known as

This theorem is sometimes referred to as the pasting lemma.

Also see

Pasting Lemma for Union of Open Sets for an analogous statement for open sets.

Sources

2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... : $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings

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1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous): Problems: Section $1: \ 5$
2000: James R. Munkres: Topology (2nd ed.): $2$: Topological Spaces and Continuous Functions: $\S 18$: Continuous Functions: Theorem $18.3$: The pasting lemma