§ Level set of a continuous function must be closed

• Let $f$ be continuous, let $L \equiv f^{-1}(y)$ be a level set. We claim $L$ is closed.
• Consider any sequence of points $s: \mathbb N \to L$. We must have $f(s_i) = y$since $s(i) \in L$. Thus, $f(s_i) = y$ for all $i$.
• By continuity, we therefore have $f(\lim s_i) = \lim f(s_i) = y$.
• Hence, $\lim s_i \in L$.
• This explains why we build Zariski the way we do: the level sets of functions must be closed. Since we wish to study polynomials, we build our topology out of the level sets of polynomials.