Continuous Mapping on Finite Union of Closed Sets

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 $C = \displaystyle \bigcup_{i=1}^n C_i$, 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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({f \restriction_{C} }\right)^{-1} \left({V}\right)\)	\(=\)	\(\displaystyle C \cap f^{-1} \left({V}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of restriction
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({ \bigcup_{i=1}^n C_i }\right) \cap f^{-1} \left({V}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of $C$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \bigcup_{i=1}^n \left({ C_i \cap f^{-1} \left({V}\right) }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Intersection Distributes over Union
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \bigcup_{i=1}^n U_i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	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.

$\blacksquare$

See also

• Continuous Mapping on Union of Opens for an analogous statement for open sets.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970): Problems: $\S 1: \ 5$