# General topology – Show that \$ Z (I_ {A}) = A \$

Problem:

Consider $$X$$ a compact Hausdorff room with $$C (X)$$ the set of continuous functions $$X$$, If $$A$$ is closed subset of $$X$$, define $$I_ {A} = {f in C (X) | f | _ {A} = 0 }$$,
and $$Z (I): = {x in X | f (x) = 0$$ for all $$f in I }$$,

$$1$$ ) Show that $$Z (I_ {A}) = A$$,

First if $$x in A$$then for everyone $$f in I_ {A}, f (x) = 0$$, so $$x in Z (I_ {A})$$, The other direction confuses me (it could be wrong). Is not it possible to have an ideal of function? $$C (X)$$ Be zero $$A$$ but also on other sets? Note appreciated. I know that $$I_ {A}$$ is an ideal that is closed in relation to the norm topology (the sup norm).