Related papers: On Borel subsets of generalized Baire spaces

It is consistent that the continuum be arbitrary large and no absolute $\kappa$-Borel set $X$ of density $\kappa$, $\aleph_1<\kappa<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large…

This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure…

We isolate several classes of stationary sets of kappa^omega and investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.

Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces $(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$ topology) for various $\theta \geq \kappa$. Let l be the number of such different…

We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel…

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{\alpha\in\kappa}b(\alpha)$ of the generalised Baire space ${}^\kappa\kappa$, where $\kappa$ is strongly inaccessible and…

Filtrations are certain transfinite sequences of topologies increasing in strength and interpolating between two given topologies $\sigma$ and $\tau$, with $\tau$ being stronger than $\sigma$. We prove general results on stabilization at…

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where…

We study the continuous reducibility of isomorphism relations in the space of regresive functions in $\kappa^\kappa$. We show for inaccessible $\kappa$, that if $\mathcal{T}$ is a theory with less than $\kappa$ non-isomorphic models of size…

We consider the following dichotomy for $\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa$: either all $R$-independent sets are of size at most $\kappa$, or there is a $\kappa$-perfect…

We prove that any suitable generalization of Laver forcing to the space $ \kappa^\kappa$, for uncountable regular $\kappa$, necessarily adds a Cohen $\kappa$-real. We also study a dichotomy and an ideal naturally related to generalized…

We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $\kappa^\kappa$ and $2^\kappa$, for regular uncountable cardinals $\kappa$.

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let {\sf BC}$_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture.…

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic…

We prove that for every uncountable cardinal $\kappa$ such that $\kappa^{<\kappa}=\kappa$, the quasi-order of embeddability on the $\kappa$-space of $\kappa$-sized graphs Borel reduces to the embeddability on the $\kappa$-space of…

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmb\Pi_\omega^0$-complete set of models. In…

We investigate generalizations of the topology of the higher Cantor space on $2^\kappa$, based on arbitrary ideals rather than the bounded ideal on $\kappa$. Our main focus is on the topology induced by the nonstationary ideal, and we call…

Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending…

We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa+1})$ has both the $\kappa$-Perfect Set Property and the $\mathcal{\vec{U}}$-Baire Property. This is a higher analogue of…

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related…