This is a follow up question to this one. If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-nor …

I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$. Furthermore, I do have a uniformly continuous functio …

Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true? $$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$ I think …

Suppose $(X,\mathcal D)$ is a precompact uniform space and $D\in \mathcal D$. there's a finite set $F\subseteq X$ such that $$D[F]=X$$ Let $\mathcal T$ be the topology on $X$ gener …

I need a proof for this proposition: If a uniformity $\mathfrak U$ on $X$ has a countable fundamental system of entourages, then it can be defined by a pseudometric on $X …

Let $\: \langle X,\hspace{-0.015 in}\delta\hspace{.005 in}\rangle \:$ be a separated proximity space. Let $\;\;\; \mu^* \: : \: 2^{\hspace{.01 in}X} \: \to \: \left[0,\hspace{-0.0 …

And what else can be said, if so? (Original math.SE post) In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniform …

I have proved that if you collapse a closed subset of a separated (Hausdorff) uniform space to a point, you get a separated uniform space. Surely such a simple result must be stan …

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well kn …

I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field $F$ which is comp …

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topolog …

Let Dom be a uniform space, and f be a continuous function from Dom to itself satisfying: (1)For all non-empty open subsets U and V of Dom, there exists a natural number n and a m …