Absolute and non-absolute $\mathcal F$-Borel spaces

TL;DR

This paper studies the complexity of $\\mathcal{F}$-Borel topological spaces, focusing on when this complexity is absolute across compactifications and introducing new representations to analyze their hierarchy.

Contribution

It establishes the absoluteness of complexity for metrizable spaces, provides conditions for absolute $\mathcal{F}_{\sigma\delta}$ spaces, and constructs a hierarchy of non-absolute $\mathcal{F}$-Borel spaces.

Findings

Complexity of metrizable spaces is absolute.

Provided a sufficient condition for spaces to be absolutely $\mathcal{F}_{\sigma\delta}$.

Constructed a hierarchy of $\mathcal{F}$-Borel spaces with non-absolute complexity.

Abstract

We investigate $\mathcal F$-Borel topological spaces. We focus on finding out how a~complexity of a~space depends on where the~space is embedded. Of a~particular interest is the~problem of determining whether a~complexity of given space $X$ is absolute (that is, the~same in every compactification of $X$). We show that the~complexity of metrizable spaces is absolute and provide a~sufficient condition for a~topological space to be absolutely $\mathcal F_{\sigma\delta}$. We then investigate the~relation between local and global complexity. To improve our understanding of $\mathcal F$-Borel spaces, we introduce different ways of representing an~$\mathcal F$-Borel set. We use these tools to construct a~hierarchy of $\mathcal F$-Borel spaces with non-absolute complexity, and to prove several other results.