An easy, non-silly example (that is perhaps more appealing than the Zariski topology to a student at the level of someone asking this question) is simply to consider the space of real-valued integrable functions on $[0,1]$ with the pseudo-norm $\|f\| = \int_0^1 |f|$. The topology generated by the balls is not Hausdorff, an explicit example of a sequence converging to two points is simply the constant sequence $f_n = 0$, which converges both to the constant $0$ function as well as the function $f(x) = 0$ for $x \in [0,1)$, $f(1) = 1$. For reasons like
An easy, non-silly example (that is perhaps more appealing than the Zariski topology to a student at the level of someone asking this question) is simply to consider the space of real-valued integrable functions on $[0,1]$ with the pseudo-norm $\|f\| = \int_0^1 |f|$. The topology generated by the balls is not Hausdorff, an explicit example of a sequence converging to two points is simply the constant sequence $f_n = 0$, which converges both to the constant $0$ function as well as the function $f(x) = 0$ for $x \in [0,1)$, $f(1) = 1$. For reasons like this, people tend to work with integrable functions up to a.e. equivalence.