Rudin's RCA $4.9$ There are  the definitions  which we need for   $4.9$ :


There is it :

I don't understand how  does the closedness of subspace $x^\bot$ follow from the fact that $x^\bot$ is precisely the set of points where the continuous function $y$ $\rightarrow$ $(x,y)$ is $0$ ?
Any help would be appreciated.

Because continuity of a function $f:X\to\Bbb R$ implies (and is equivalent to) $f^{-1}(A)$ being closed in $X$ for every $A$ closed in $\Bbb R$. In particular, $f^{-1}(0)$ is closed as $\{0\}$ is closed.
The function $f:y\mapsto(x,y)$ is continuous, so $f^{-1}(0)=x^{\perp}$ must be closed.
