Extension of Extension of Mapping is Extension

Theorem

Let $A, B, C, S$ be sets such that $A \subseteq B \subseteq C$.

Let $f: A \to S$, $g: B \to S$ and $h: C \to S$ be mappings such that:

$g$ is an extension of $f$ to $B$

$h$ is an extension of $g$ to $C$.

Then $h$ is an extension of $f$ to $C$.

Proof

By definition of extension:

$\forall x \in A: \map f x = \map g x$

and:

$\forall x \in B: \map g x = \map h x$

and so:

$\forall x \in A: \map g x = \map h x$

from which it follows that:

$\forall x \in A: \map f x = \map h x$

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions: Exercise $3$