4 - The complex topology

Summary

We have now defined schemes locally of finite type over ℂ. Let (X, O) be such a scheme. The scheme (X, O) is a ringed space, and in particular X is a topological space. The topology is a little strange, as we saw in Remark 3.1.10: the space X is almost never Hausdorff, even worse, not every point in X is closed. It is natural enough to look at the subset of closed points of X. We can give it the subspace topology, in which case we denote it by Max(X). But the set Max(X) turns out to have another topology, called the complex topology. We denote this topological space X^an. As sets of points Max(X) and X^an agree; only the topologies differ.

Synopsis of the main results

We now briefly summarize the main results of this chapter. The notation is as in the paragraph above. This chapter will prove:

4.1.1. Let (X, O) be a scheme locally of finite type over ℂ. Then the natural map λ_X : X^an → X is continuous. This means the following: when we forget the topology X^an is simply the set of all closed points in X, and there is an obvious inclusion map λ_X : X^an → X. We are asserting that, if we give X its Zariski topology and X^an its complex topology, then the map λ_X is continuous.