12 problems

Let $X$ be a topological space, and let $\mathcal F[X]$ denote its finite-set hyperspace. A space is discretely selective if it has the selection property referred to in the preced…

Let $X$ be a topological space, and let $CL(X)$ denote the set of all nonempty closed subsets of $X$, endowed with the Vietoris topology $\tau_V$. The quasi-metrizability conjectur…

Let $X$ be a space, and let $CL(X)$ denote the hyperspace of nonempty closed subsets of $X$ with Vietoris topology $fmathbb{V}$. Conjecture 2. $(CL(X), \mathbb{V})$ is quasi-metriz…

Let $X$ be a space. Write $CL(X)$ for the hyperspace of nonempty closed subsets of $X$, equipped with the Vietoris topology $fmathbb{V}$. A topological sum is denoted by…

Isbell's conjecture. The structures $\exp\mathcal{U}_{1}$ and $\exp\mathcal{U}_{2}$ induce different topologies on $H\left(X\right)$.

Let $(X,d)$ be a metric space. The normality–separability conjecture. If the Wijsman hyperspace $(2^X,\mathscr T_{W(d)})$ is normal, then $(X,d)$ is separable. This conjecture woul…

For a map $f$, write $\operatorname{Per}(2^f)$ for the set of periods of the induced map $2^f$. Let $P\subset\mathbb N$ be finite. Periodicity-set conjecture. There exists a map…

Let $X$ be a dendrite, let $f:X\to X$ be a continuous dendrite map, and let $2^f:2^X\to 2^X$ be the induced map on the hyperspace of nonempty compact connected subsets of $X$. A po…

Component-diameter conjecture. If $C$ is a transitive point for $f_*$ on $2^X$, then

Let $(X,\tau)$ be a topological space, and let $PR(X)=[X]^{<\omega}$ be the set of finite subsets of $X$ equipped with the Pixley–Roy topology, whose basic open sets are … for…

Let $\ell^2_+$ be the positive cone of the separable Hilbert space $\ell^2$. A subset $B\subset\ell^2_+$ is $\mathbb B$-convex if, for all $x,y\in B$ and $t\in[0,1]$, one has…

For an infinite cardinal $\tau$, let $\operatorname{cc}(\mathbb R^\tau)$ denote the hyperspace of compact subsets of the product space $\mathbb R^\tau$. Hyperspace homeomorphism co…