continuity is preserved when codomain is extended

Theorem 1.

Suppose $X\mathrm{,}Y$ are topological space andlet $Z\mathrm{\subseteq }Y$ be equipped with the subspace topology.If

$$f:X\to Z$$

is continuous, then

$$f:X\to Y$$

is continuous.

Proof.

Let $U\subseteq Y$ be an open set. Then

$f^{-1}(U)$$=$$\mathrm{\{}x\in X:f(x)\in U\}$

$=$$\mathrm{\{}x\in X:f(x)\in U\cap Z\}$

$=$$f^{-1}(U\cap Z).$

Since $U\cap Z$ is open in $Z$, $f^{-1}(U)$ is open in $X$.∎