Complexities and Representations of F-Borel Spaces

TL;DR

This paper explores the complexity of F-Borel spaces across different compactifications, introduces a representation theory for these sets, and characterizes the complexities of hereditarily Lindelöf spaces, revealing their absolute nature.

Contribution

It develops a new theory of representations for F-Borel sets and applies it to analyze the complexity invariance in various compactifications, providing alternative proofs and new insights.

Findings

Spaces can have different complexities in different compactifications.

Hereditarily Lindelöf spaces have absolute complexity across all compactifications.

Constructs spaces with non-absolute additive complexity.

Abstract

We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifcations. We provide a simple proof of the fact that a space can have arbitrarily many different complexities in different compactifications. We also develop a theory of representations of $\mathcal F$-Borel sets, and show how to apply this theory to prove that the complexity of hereditarily Lindel\"of spaces is absolute (that is, it is the same in every compactification). We use these representations to characterize the complexities attainable by a specific class of topological spaces. This provides an alternative proof of the first result, and implies the existence of a space with non-absolute additive complexity. We discuss the method used by Talagrand to construct the first example of a space with non-absolute complexity, hopefully providing an explanation which is more accessible than the original one. We also discuss the relation of complexity and local complexity, and show how to construct amalgamation-like compactifications.