§ Why the zero set of a continuous function must be a closed set

• Consider the set of points $Z = f^{-1}(0)$ for some function $f: X \to \mathbb R$.
• Suppose we can talk about sequences or limits in $X$.
• Thus, if $f$ is continuous, then we must have $f(\lim x_i) = \lim f(x_i)$.
• Now consider a limit point $l$ of the set $Z$ with sequence $l_i$ (that is, $\lim l_i = l$). Then we have $f(l) = f(\lim l_i) = \lim f(l_i) = \lim 0 = 0$. Thus, $f(l) = 0$.
• This means that the set $Z$ contains $l$, since $Z$ contains all pre-images of zero. Thus, the set $Z$ is closed.
• This implies that the zero set of a continuous function must be a closed set.
• This also motivates zariski; we want a topology that captures polynomial behaviour. Well, then the closed sets must be the zero sets of polynomials!