The Zariski topology is part of the basic structure of varieties and schemes.  Unlike other,
fancier Grothendieck topologies, it is actually a topology, defined by subsets of the variety/scheme, and so gives rise to the notion of closed, as well as of open, subset.  The closed subsets are the algebraic subsets (in the variety case) or (the spaces underlying) the closed subschemes (in the scheme case), which are what the study of algebraic geometry is to a large extent about.  

If you look at Hartshorne, Chapters IV and V, you will find a *lot* of geometry of curves and surfaces (as well as some geometry of higher dimensional varieties too), all of it treated without recourse to any topology other than the Zariski topology.