§ Level set of a continuous function must be closed
- Let be continuous, let be a level set. We claim is closed.
- Consider any sequence of points . We must have since . Thus, for all .
- By continuity, we therefore have .
- Hence, .
- 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.