If you read Serre's FAC you will notice the definition of algebraic variety X (section 34 page 40), http://achinger.impan.pl/fac/fac.pdf
axiom 2: $\Delta= \{(x,x) \mid x \in X\} \subset X \times X$ is closed in the induced topology of X $\times$ X.
In the product topology (which looks like the induced topology... but I may be wrong here) $\Delta$ closed iff X Hausdorff
He later uses the Zariski topology on X. But we know the Zariski topology is not Hausdorff.
What is wrong with my interpretation of the induced topology?
This is because the topology on $X \times X$ is not the product topology when we use the Zariski topology. You can think of $\mathbb A^2 = \mathbb A^1 \times \mathbb A^1$ for convince yourself of this fact.