A survey of cardinality bounds on homogeneous topological spaces
Nathan Carlson

TL;DR
This survey compiles and analyzes decades of results on bounds for the cardinality of homogeneous topological spaces, introduces new bounds, and discusses proof techniques and improvements.
Contribution
It provides a comprehensive overview of known bounds, introduces new cardinality bounds for homogeneous spaces, and discusses proof methods and recent generalizations.
Findings
Includes bounds like |X| ≤ 2^{πw(X)} and improvements for various classes.
Introduces new bounds involving πnχ(X) and qψ(X).
Provides a table of strongest known bounds and some proofs.
Abstract
In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen's Theorem, which states if is a power homogeneous Hausdorff space, and its improvements and for spaces with the same properties. We also discuss de la Vega's Theorem, which states that if is a homogeneous compactum, as well as its recent improvements and generalizations to other settings. This reference document also includes a table of strongest known cardinality bounds on spaces with homogeneous-like properties. The author has chosen to give some proofs if they exhibit typical or fundamental proof techniques. Finally, a few new results are given, notably (1)…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Fuzzy and Soft Set Theory
