Zariski topology

In this topology, named after Oscar Zariski, the closed sets are the sets consisting of the mutual zeros of a finite set of polynomial equations.

This definitions indicates the kind of space that can be given a Zariski topology: for example we define the Zariski topology on a n-dimensional vector space F^n over a field F, using the definition above. That this definition yields a true topology is easily verified.

It follows easily that homomorphisms are continuous and so the Zariski topology given to some finite-dimensional vector space doesn't depend on a specific basis chosen.

From here one can generalise the definition of Zariski topology to infinite-dimensional vector spaces, projective spaces, and subsets of these.