  In this paper we define the algebraic sets and the ideal of points for
bijective skew PBW extensions with coefficients in left Noetherian domains.
Some properties of affine algebraic sets of commutative algebraic geometry will
be extended, in particular, a Zariski topology will be constructed. Assuming
additionally that the extension is quasi-commutative with polynomial center and
the ring of coefficients is an algebraically closed field, we will prove an
adapted version of the Hilbert's Nullstellensatz theorem that covers the
classical one. The Gr\"obner bases of skew PBW extensions will be used for
defining the algebraic sets and for proving the main theorem. Many key algebras
and rings coming from mathematical physics and non-commutative algebraic
geometry are skew PBW extensions.
