# 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

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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