# Continuous Mapping on Finite Union of Closed Sets

## Contents

## Theorem

Let $T = \left({X, \tau}\right)$ and $S = \left({Y,\sigma}\right)$ be topological spaces.

For all $i \in \left\{{1, 2, \ldots, n}\right\}$, let $C_i$ be closed in $T$.

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

Then $f$ is continuous on $C = \displaystyle \bigcup_{i \mathop = 1}^n C_i$, i.e., $f \restriction_C$ is continuous.

If $\left\{{C_i}\right\}$ is infinite, the result does not necessarily hold.

## Proof

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

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

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

Therefore, we can compute:

\(\displaystyle \left({f \restriction_{C} }\right)^{-1} \left({V}\right)\) | \(=\) | \(\displaystyle C \cap f^{-1} \left({V}\right)\) | Definition of restriction | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({ \bigcup_{i \mathop = 1}^n C_i }\right) \cap f^{-1} \left({V}\right)\) | Definition of $C$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \bigcup_{i \mathop = 1}^n \left({ C_i \cap f^{-1} \left({V}\right) }\right)\) | Intersection Distributes over Union | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \bigcup_{i \mathop = 1}^n U_i\) | Definition of $U_i$ |

That is, $U = \left({f \restriction_{C} }\right)^{-1} \left({V}\right)$ is the intersection 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

- Continuous Mapping on Union of Open Sets for an analogous statement for open sets.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*: Problems: $\S 1: \ 5$ - 2000: James R. Munkres:
*Topology*(2nd ed.): $2$: Topological Spaces and Continuous Functions: $\S 18$: Continuous Functions: Theorem $18.3$: The pasting lemma