arXiv Algebraic Geometry
arXiv Algebraic Geometry
Preprint
4 years ago

Compactifications of topological spaces via commutative algebra. (arXiv:1910.09884v3 [math.AC] UPDATED)

A. Tarizadeh, M. R. Rezaee
In this paper, using commutative algebra, then new and significant advances on the compactifications of topological spaces, specially on the Alexandroff and Stone-\v{C}ech compactifications have been made. First, the Alexandroff compactification of a discrete space is constructed by a new and interesting approach. Then it is proved that the minimal spectrum of the direct product of a family of integral domains and also the maximal spectrum of the direct product of a family of local rings both indexed by a set are the Stone-\v{C}ech compactification of the discrete space . These results improve all of the previous constructions of the Stone-\v{C}ech compactification of a discrete space. Some applications of this study are given. Specially and surprisingly, it is shown that the Stone-\v{C}ech compactification of an arbitrary topological space is obtained from the Stone-\v{C}ech compactification of the discrete space by passing to a its appropriate quotient. We give a new and simple way to construct ultra-rings, and then this new approach is used to obtain non-trivial results on the Stone-\v{C}ech compactification.

Publisher URL: http://arxiv.org/abs/1910.09884

DOI: arXiv:1910.09884v3

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