Last time, we have seen that every completely metrizable space X has an ideal model, that is, that X can be embedded into an ideal domain Y in such a way that we can equate X with the subspace of maximal elements of Y.
We have also seen the converse to that: if X is a metrizable space with an ideal model, then X is completely metrizable.
But we had skipped an essential ingredient: that the set X of maximal elements of an ideal model Y is a Gδ subset of Y. This is true, but complicated. As I have already said last time, we shall do something slightly simpler. See the full post.