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A quasi-projective algebraic set is a locally closed subset of projective space in the Zariski topology: it is an open subset of a closed algebraic subset. Equivalently it is the intersection of a Zariski-open subset and a Zariski-closed subset of the same projective space. Reducibility is allowed; an irreducible such space is a quasi-projective variety.

Every projective variety is quasi-projective, by taking the whole closed subset as the open part. Every affine variety is also quasi-projective through its usual affine chart in projective space. Quasi-projective does not mean affine: the projective line itself is quasi-projective but not affine, since all its global regular functions are constant whereas it has more than one point. Its complement of one point is an affine line.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

A Zariski cover of a scheme is a family of open immersions whose images cover its underlying topological space. Such families define the Zariski topology as a Grothendieck topology on the category of schemes; a cover need not be finite.

A subset of an algebraic variety is Zariski dense when its closure in the Zariski topology is the whole variety. For an affine variety, this is equivalent to saying that every regular function vanishing on the subset vanishes on the entire variety.