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Results tagged with descriptive-set-theory
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user 43266
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
4
votes
Picking a real for every non-empty open set in $\mathbb{R}$
With the axiom of choice:
if $\kappa$ is an infinite cardinal, then any collection of (at most) $\kappa$ sets, each of cardinality (at least) $\kappa$, has an injective choice function. In this case $ …
- 13.4k
6
votes
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
Various questions about partitioning a topological space $X$ into homeomorphic copies of a topological space $Y$ are discussed by Paul Bankston and Richard J. McGovern, Topological partitions, General …
- 13.4k
2
votes
Accepted
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
A set $B\subseteq\mathbb R$ is a Bernstein set if $B$ contains no nonempty perfect set while having nonempty intersection with every nonempty perfect set; in other words, if neither $B$ nor $\mathbb R …
- 13.4k
18
votes
Accepted
Avoiding countable subgroups of general uncountable groups
Counterexample for Problem 1: According to this answer Saharon Shelah constructed a "Jónsson group" of order $\aleph_1,$ i.e., an uncountable group $G$ in which every proper subgroup is countable. Cho …
- 13.4k