11 problems

An analytic equivalence relation is an equivalence relation on a Polish space whose graph is analytic. An analytic equivalence relation is universal if every analytic equivalence r…

Let $\kappa$ be an infinite cardinal, let $\omega$ denote the discrete space of natural numbers, and let $X$ be a connected manifold. Countable dense homogeneity extension conjectu…

Let $X$ be a perfect Polish space, meaning a separable completely metrizable space with no isolated points. Let $cmathsf{BA}(X)$ denote the Baumgartner axiom parametrized by $X$, a…

Let $cmathsf{BA}$ denote Baumgartner's axiom for the real line, and let $cmathsf{BA}(X)$ be its parametrized version for a topological space $X$. For a positive integer $n$, write…

Let $({\bf Y},d)$ be one of the metric spaces in cases ${\bf (A)}$--${\bf (C)}$ of Section 2.3, and let $S$ be the corresponding approximating dense substructure. Let ${\bf Y}_L$ d…

Let $({\bf Y},d)$ be one of the metric spaces in cases ${\bf (A)}$--${\bf (C)}$ of Section 2.3, and let $S$ be the corresponding approximating dense substructure. Let ${\bf Y}_L$ d…

Let $X$ be a Hausdorff topological group. Baire–banalytic characterization conjecture. The group $X$ is Polish if and only if $X$ is a Baire space and a banalytic space. This conje…

Let $(X,\rho)$ be an infinite Polish ultrametric space. A map $\varphi:X\rightarrow X$ is locally contractive if, for every $x\in X$, there is a neighborhood on which distances fro…

Gao's conjecture. Isometry between zero-dimensional Polish metric spaces is Borel bireducible with any complete orbit equivalence relation.

Let $X$ be an analytic space, $Y$ a separable metrizable space, $1<n<m<\omega$, and $f\in\Sigma_{m,n}(X,Y)$. Topology-refinement reformulation of the strong Jayne–Rogers conjecture…

Dichotomy conjecture. There is a dichotomy: either the model is similar to categorical theories, or there are “many complicated models”.