We are told that the Zariski topology is not Hausdorff, but I have rarely seen "concrete examples" of the dramatic failures this can induce.
A concrete example I have in mind:
For example, one way I know to show that the Zariski topolgy is weird is as follows:
We claim that on $\mathbb R$ equipped with the zariski topology, two points $x, y \in \mathbb R, x \not= y$ cannot be separated by open sets $X, Y \subseteq \mathbb R$ such that $x \in X, y \in Y, X \cap Y = \emptyset$.
The idea is that since the closed sets induced by the Zariski topology are of the form $\{ x \in \mathbb R : \forall i \in I, f_i(x) = 0, f_i(x) \in \mathbb R[X] \}$
Hence the open sets will be of the form $\{x \in \mathbb R: \exists i \in I, f_i(x) \neq 0, f_i(x) \in \mathbb R[X]$ }.
However, in general, polynomials are zero at finitely many locations. Hence, the sets where polynomials are not zero are extremely large. Therefore, it is hard to separate two points with open sets, since the non-zero sets of polynomials will likely intersect in a large region of space.
I don't understand this handway example very well either: I would like to see a formal proof of this "fact" that I have learnt from the folklore.
Examples of dramatic failures I would like:
- concrete functions becoming continuous that we do not think are continuous.
- concrete functions becoming discontinuous that we do not think are discontinuous (I think this cannot happen, since the Zariski topology is very coarse. However, I don't have a good intuition for the topology, so if somoene can correct me on this, I would be glad)
- A concrete sequence having multiple limits