A-spectral spaces. (Q1426498)

A topological space \(X\) is said to be spectral if and only if it satisfies the following four conditions: (1) It is sober; (2) it is quasi-compact; (3) the quasi-compact open subsets form a basis for the topology of \(X\); (4) the family of quasi-compact open subsets of \(X\) is closed under finite intersections. The given conditions describe those topological spaces that are homeomorphic to the prime spectrum of a commutative ring with identity endowed with the Zariski topology. In the article under review the authors characterize the topological spaces \(X\) whose Alexandroff extension or one-point-compactification \(\widetilde{X}\) is spectral. These spaces are called \(A\)-spectral. (Here, as usual, for a topological space \(X\), the one-point-extension \(\widetilde{X}=X\cup\{\infty\}\) is equipped with the topology whose members are the open subsets of \(X\) and all subsets \(U\) of \(\widetilde X\) such that \(\widetilde{X}\setminus U\) is a closed quasi-compact subset of \(X\).)