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 continuousmapping ${f|}_A:A\to Y$, where $A$ is given the subspace topologyfrom $X$.

Proof. We need to show that for any open set $V\subset Y$, wecan 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 (seethis 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 $X$, andour claim follows. $\mathrm{\square }$