Quasi-Compactness in Infinite Dimension

TL;DR

This paper characterizes quasi-compactness in infinite-dimensional affine spaces and inverse limits of prime spectra, providing criteria and examples of non-quasi-compact cases.

Contribution

It offers comprehensive criteria for quasi-compactness in infinite-dimensional settings and presents an example of a non-quasi-compact affine space.

Findings

Weak stability, retro-compactness, and cylinder sets are equivalent criteria for quasi-compactness.

An example of a non-quasi-compact affine space is constructed.

Extensive characterizations for quasi-compactness in infinite-dimensional contexts are provided.

Abstract

We give extensive characterizations for an open subset of an affine space of arbitrary dimension, resp. of an inverse limit of prime spectra to be quasi-compact. Among other things weak stability, retro-compactness, and cylinder sets provide equivalent criteria in both settings. We also exhibit an example of a non-quasi-compact affine space.