A Universe of Sorts

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.

§ Level Set of a Continuous Function Must Be Closed