On the cardinality of Hausdorff spaces

TL;DR

This paper develops a unified framework for cardinality inequalities in Hausdorff spaces, extending known results and analyzing chains of spaces, with implications for understanding their size limitations.

Contribution

It introduces a generalization connecting two main streams of cardinality inequalities for Hausdorff spaces, extending existing results and analyzing chains of spaces.

Findings

Unified framework for cardinality inequalities

Extension of inequalities by Bella and Cammaroto

Analysis of chains of spaces satisfying these inequalities

Abstract

A common generalization for two of the main streams of cardinality inequalities is developed; each stream derives from the famous inequality established by A.V. Arhangel'ski\u{\i} in 1969 for Hausdorff spaces. At the end of one stream is the recent inequality by Bella and at the end of the second stream is the 1988 inequality by Bella and Cammaroto. This generalization is extended and used to analyze a result containing an increasing chain of spaces that satisfies the same cardinality inequality. The paper is concluded with some open problems.