# continuity is preserved when codomain is extended

###### Theorem 1.

Suppose $X,Y$ are topological space and let $Z\subseteq Y$ be equipped with the subspace topology. If

 $f\colon X\to Z$

is continuous, then

 $f\colon X\to Y$

is continuous.

###### Proof.

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

 $\displaystyle f^{-1}(U)$ $\displaystyle=$ $\displaystyle\{x\in X:f(x)\in U\}$ $\displaystyle=$ $\displaystyle\{x\in X:f(x)\in U\cap Z\}$ $\displaystyle=$ $\displaystyle f^{-1}(U\cap Z).$

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

Title continuity is preserved when codomain is extended ContinuityIsPreservedWhenCodomainIsExtended 2013-03-22 15:17:14 2013-03-22 15:17:14 matte (1858) matte (1858) 9 matte (1858) Theorem msc 54C05 IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous