A Descriptive Set-theoretic Analysis Of Path-connectedness

Date

2025-8-6

Author

Uyar, Yusuf

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In this thesis, we analyze the complexity of path-connectedness and some other topological notions related to connectedness in R^2 and R^3 from the point of view of descriptive set theory. More specifically, following Debs, Saint Raymond and Becker, we survey the maximal descriptive complexity of the collection of subsets satisfying certain connectedness properties inside hyperspaces on Polish spaces and then, give examples of subspaces of R^n for n=2 and n=3 in which these complexity bounds are realized in various cases. In addition, we also examine the equivalence relation of path-connectedness on Polish subspaces of R^2 using tools of Borel complexity theory. More specifically, we prove that the path-connectedness relation of a Polish subspace of R^2 is an essentially countable Borel equivalence relation. We also show that the path-connectedness relation of the Knaster continuum is a non-smooth essentially hyperfinite Borel equivalence relation.

Subject Keywords

Descriptive complexity, Path-connected, Connected, Borel reducible, Borel equivalence relations

URI

https://hdl.handle.net/11511/115508

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Graduate School of Natural and Applied Sciences, Thesis