§ Why the zero set of a continuous function must be a closed set
- Consider the set of points for some function .
- Suppose we can talk about sequences or limits in .
- Thus, if is continuous, then we must have .
- Now consider a limit point of the set with sequence (that is, ). Then we have . Thus, .
- This means that the set contains , since contains all pre-images of zero. Thus, the set 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!