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 Xan. As sets of points Max(X) and Xan 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 λX : Xan → X is continuous. This means the following: when we forget the topology Xan is simply the set of all closed points in X, and there is an obvious inclusion map λX : Xan → X. We are asserting that, if we give X its Zariski topology and Xan its complex topology, then the map λX is continuous.