While the Zariski has its limitations, it amazes me how well it does work. A few brief points in its defense:

1. It's easy to define. In the classical case of affine spaces over a field, it's the weakest topology for which points are closed and polynomials are continuous.

2. It can be used to give precise meaning to the word "generic" or "general", as in "a general matrix is diagonalizable, therefore to prove Cayley-Hamilton it suffices..."

3. For coherent sheaves, it's the right topology; cohomology works as expected. At a more sophisticated level, cohomology is upper semicontinuous in the Zariski topology, and this is very important for many arguments.

So for the younger generation out there who are thinking of doing away with it: please don't!

While the Zariski topology has its limitations, it amazes me how well it does work. A few brief points in its defense:

1. It's easy to define. In the classical case of affine spaces over a field, it's the weakest topology for which points are closed and polynomials are continuous.

2. It can be used to give precise meaning to the word "generic" or "general", as in "a general matrix is diagonalizable, therefore to prove Cayley-Hamilton it suffices..."

3. For coherent sheaves, it's the right topology; cohomology works as expected. At a more sophisticated level, cohomology is upper semicontinuous in the Zariski topology, and this is very important for many arguments.

So for the younger generation out there who are thinking of doing away with it: please don't!