This definitions indicates the kind of space that can be given a Zariski topology: for example we define the Zariski topology on a ndimensional 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 finitedimensional vector space doesn't depend on a specific basis chosen.
From here one can generalise the definition of Zariski topology to infinitedimensional vector spaces, projective spaces, and subsets of these.
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