Algebraic geometry via universal domain

In the modern approach using scheme theory, Weil's approach is
encapsulated in the following observation. Given a finite type scheme X
over K and a point x in X, the residue field K(x) of the local ring
O\_{X,x} is a finitely generated field extension of K, hence it admits an
embedding K(x) --> Omega (sorry I seem to have a problem with encoding and cannot get Omega to work properly in this answer box). This corresponds in turn to a morphism Spec(Omega)-->X with image x. In other words, the map X(Omega) --> |X| from the Omega-points of X to the set of points of the scheme X is **surjective**.

You can then take the preimage of the Zariski topology on |X| to X(Omega) and get a topology on X(Omega). What Weil did was basically to work directly with this topology.

I would recommend learning directly the language of varieties and then schemes, and not spend any time with Weil's approach until you can make sense of the previous paragraphs - and then you will see that this approach, while historically important, is completely superseded by the language of schemes.
If you read the very next sentence of the Wikipedia article, it elaborates that Zariski did not like having two points that were topologically indistinguishable. This seems quite a reasonable assumption to put on spaces: if there is literally no topological way to distinguish two points, how on Earth can we say they are the same? We need to be leaving set theory, not going more towards it!
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