47 problems

Let $2^\omega$ be the Cantor group under coordinatewise addition, and call $X\subseteq 2^\omega$ strongly meager if, for every measure zero set $H$, one has … Dual Borel Conjecture…

Let $\lambda$ and $\kappa$ be cardinals, and let ${\cal K}{\cal L}(\lambda,\kappa)$ denote the Kalikow property: there is a sequence of functions $\langle F_n:n<\omega\rangle$ with…

Let ${\cal I}$ and ${\cal J}$ be two of the ideals considered in the paper such that ${\cal I}\not\subseteq {\cal J}$. Covering-number separation conjecture. It is consistent that…

Countable-support iteration conjecture. The conclusion of Theorem should hold for the countable support iteration of the partial order ${\Bbb P}({\cal I})$.

Let $i$ and $j$ be integers with $i\geq j\geq 2$, let $\mathbb{B}$ be a Boolean algebra, and let $n$ be the least integer greater than or equal to $i/(j-1)$. The quantities…

A first countable (Tychonov) space is a topological space in which every point has a countable local base. A point-countable $\pi$-base is a $\pi$-base such that each point belongs…

Let $X$ be a compact homogeneous space of countable tightness, meaning that $t(X)=\aleph_0$: whenever $A\subseteq X$ and $x\in\overline{A}$, there is a countable $B\subseteq A$ suc…

Finite-power strong measure zero conjecture. If $X$ has strong measure zero and $|X|<\mathfrak{b}$, then all finite powers of $X$ have strong measure zero.

Let $C_p(X)$ denote the space of continuous real-valued functions on a set of reals $X$, with the topology of pointwise convergence. Let $\mathfrak{rez}$ be the least cardinality o…

The distinctness conjecture. If $\ne$, then

Let $G$ be a compact topological group, and let … be its density spectrum of dense subgroups. Peng's conjecture. Every compact group $G$ satisfies … This asserts that every compact…

Let $\textsf{non}^*(\mathcal{ED}_{\textsf{fin}})$ denote the corresponding cardinal invariant for the ideal of eventually different functions, and let…

Fremlin's conjecture. $\operatorname{cov}(\mathcal{N})$ has uncountable cofinality.

Let $\lambda$ be a cardinal with $\bf{\delta^1_4}\leq\lambda\leq\bf{\delta^1_6}$, and suppose $\operatorname{cof}(\lambda)>\omega$. A cardinal is $\bf{\Gamma}$-unreachable when the…

Let $\bf{\Gamma}$ be a pointclass, and let $\kappa$ be $\bf{\Gamma}$-reachable when there is a sequence of distinct $\bf{\Gamma}$-sets of length $\kappa$; otherwise, $\kappa$ is…

Let $G$ be a compact topological group. A subgroup $H\leq G$ is dense if its closure in $G$ is $G$, and $d(H)$ denotes the density character of $H$, while $w(G)$ denotes the weight…

Let $X$ be a topological space, let $o(X)$ denote its relevant cardinal invariant, and let $\mathscr U_X$ be the family of uniform topologies on $C(X)$. Under the set-theoretic ass…

Invariance conjecture. For every compact Polish space $X$ with

Separability conjecture. The space $X$ is separable.

For a Polish group $\mathbb{G}$, define … Also let $\operatorname{non}(\operatorname{Smz}(\mathbb{G}))$ be the least cardinality of a subset of $\mathbb{G}$ that is not of strong m…

Arbitrariness conjecture. The cardinal number $\kappa(\mathscr T,\mathcal I)$ may be any cardinal number.

Let $X$ be a space, let $Z(X)$ denote the family of zero-sets in $X$, and let $L(X)$ be the set of all finite linear combinations of characteristic functions of members of $Z(X)$.…

Proper destruction conjecture for MAD families. Every MAD family can be destroyed by a proper forcing that does not add dominating or unsplit reals.

Hrušák's conjecture. Either

A function $U:[\omega_{1}]^{2}\longrightarrow\omega$ is $(1,\omega)$-weakly universal if for every function $F:[\omega_{1}]^{2}\longrightarrow\omega$ there are an injective functio…