# restriction of a continuous mapping is continuous

Theorem Suppose $X$ and $Y$ are topological spaces, and suppose $f:X\to Y$ is a continuous function. For a subset $A\subset X$, the restriction (http://planetmath.org/RestrictionOfAFunction) of $f$ to $A$ (that is $f|_{A}$) is a continuous mapping $f|_{A}:A\to Y$, where $A$ is given the subspace topology from $X$.

Proof. We need to show that for any open set $V\subset Y$, we can write $(f|_{A})^{-1}(V)=A\cap U$ for some set $U$ that is open in $X$. However, by the properties of the inverse image (see this page (http://planetmath.org/InverseImage)), we have for any open set $V\subset Y$,

 $(f|_{A})^{-1}(V)=A\cap f^{-1}(V).$

Since $f:X\to Y$ is continuous, $f^{-1}(V)$ is open in $X$, and our claim follows. $\Box$

Title restriction of a continuous mapping is continuous RestrictionOfAContinuousMappingIsContinuous 2013-03-22 13:55:53 2013-03-22 13:55:53 matte (1858) matte (1858) 6 matte (1858) Theorem msc 54C05 msc 26A15