# If $f\penalty 0\@m{}M\mskip 2.0mu \mathpunct{}\nonscript\mskip-3.0mu {:}\mskip 6.0% mu plus 1.0mu X\to Y$ is continuous then $f\penalty 0\@m{}M\mskip 2.0mu \mathpunct{}\nonscript\mskip-3.0mu {:}\mskip 6.0% mu plus 1.0mu X\to f(X)$ is continuous

###### Theorem 1.

Suppose $X,Y$ are topological spaces and $f\colon X\to Y$ is a continuous function. Then $f\colon X\to f(X)$ is continuous when $f(X)$ is equipped with the subspace topology.

###### Proof.

Let us first note that using a property on this page (http://planetmath.org/InverseImage), we have

 $X=f^{-1}f(X).$

For the proof, suppose that $A$ is open in $f(X)$, that is, $A=U\cap f(X)$ for some open set $U\subset Y$. From the properties of the inverse image, we have

 $f^{-1}(A)=f^{-1}(U)\cap f^{-1}(f(X))=f^{-1}(U)$

so $f^{-1}(A)$ is open in $X$. ∎

Title If $f\penalty 0\@m{}M\mskip 2.0mu \mathpunct{}\nonscript\mskip-3.0mu {:}\mskip 6.0% mu plus 1.0mu X\to Y$ is continuous then $f\penalty 0\@m{}M\mskip 2.0mu \mathpunct{}\nonscript\mskip-3.0mu {:}\mskip 6.0% mu plus 1.0mu X\to f(X)$ is continuous IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous 2013-03-22 15:16:28 2013-03-22 15:16:28 matte (1858) matte (1858) 6 matte (1858) Theorem msc 26A15 msc 54C05 ContinuityIsPreservedWhenCodomainIsExtended