continuity is preserved when codomain is extended

Theorem 1.

Suppose $X\mathrm{,}Y$ are topological space and let $Z\mathrm{\subseteq }Y$ 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 $X$. ∎

Title	continuity is preserved when codomain is extended
Canonical name	ContinuityIsPreservedWhenCodomainIsExtended
Date of creation	2013-03-22 15:17:14
Last modified on	2013-03-22 15:17:14
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	9
Author	matte (1858)
Entry type	Theorem
Classification	msc 54C05
Related topic	IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous