The paper deals with developing classical scheme theory. Namely, the authors consider scheme-theoretical background subject – relations between spectralization and topology.
The authors show that for a field \(K\) and \(n\geq 1\), the soberification \(\mathcal{S}(\mathbb{A}^n(K))\) of the affine \(n\)-space \(\mathbb{A}^n(K)\) over \(K\) is homeomorphic to its spectralization \(\mathcal{BS}(\mathbb{A}^n(K))\), and it can be embedded into the spectrum \(\mathrm{Spec}(K[X_1,\dots,X_n])\). Moreover, if the field \(K\) is algebraically closed, then there are homeomorphisms \(\mathcal{S}(\mathbb{A}^n(K)) \approx\mathcal{BS}(\mathbb{A}^n(K))\approx \mathrm{Spec}(K[X_1,\dots,X_n])\). They also show that for a space \(X\), the subspace \(z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X))\) of prime z-ideals of the ring \(C(X)\) of real-valued continuous functions on \(X\) is homeomorphic to the space \(z\mathcal{SR}(X)\) of prime \(z\)-filters with an appropriate topology and there is a homeomorphism \(\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))\) provided \(X\) is perfectly normal. (Notions of Sober space and Soberification deals with space improving. They are defined at pages 514 and 515 resp.)
The paper is self-contained elementary and easy for reader.