9 problems

Let $X$ and $Y$ be uniform spaces, let $f:X\rightarrow Y$ be a cover, and suppose that $Y$ has uniform dimension at most $n$. Uniform dimension invariance conjecture. The spaces…

Let $X$ be a coverable uniform space with uniform dimension $u\dim X\leq n$, and let $\widetilde{X}$ denote its generalized universal cover. Uniform dimension conjecture. The unifo…

A space is locally fine when its uniformity is locally fine, and a space is subfine when it is uniformly homeomorphic to a subspace of a fine space. Isbell's conjecture. Every loca…

Let $\{a,b\}$ be the discrete topological space with two points, and let $\mathbb R$ be the real line with its usual metric. For a class $P$ of morphisms, write $P^{lr}$ for its do…

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, let $U(X)$ denote the ring of real-valued uniformly continuous functions on $X$, and let $B(F,\varepsilon)$ be the $\varepsilon$-enlargement of $F$ w…

Let $(X,d)$ be a metric space. Write $U(X)$ for the ring of real-valued uniformly continuous functions on $X$. A subset $A\subset X$ is Bourbaki-bounded if every uniformly continuo…

Let $X$ be a metrizable space, and let $X^{\prime}$ denote the set of all non-isolated points of $X$. Suppose that the finest uniformity on $X$ has an $\omega^\omega$-base of entou…

A metrizable uniform space is residually finite-dimensional if it satisfies the residual finite-dimensionality hypothesis used above. Residual finite-dimensionality conjecture. The…