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Complexities and representations of $\mathcal F$-Borel spaces

Volume 540 / 2019

Vojtěch Kovařík Dissertationes Mathematicae 540 (2019), 1-69 MSC: Primary 54H05; Secondary 54G20. DOI: 10.4064/dm794-2-2019 Published online: 27 June 2019

Abstract

We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifications. 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 Lindelöf spaces is absolute (that 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 between complexity and local complexity, and show how to construct amalgamation-like compactifications.

Authors

  • Vojtěch KovaříkCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    Praha 8, 186 00, Czech Republic
    and
    Czech Technical University in Prague
    Faculty of Electrical Engineering
    Department of Computer Science
    Karlovo náměstí 13
    Praha 2, Czech Republic
    e-mail

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