In the mathematical field of topology, a uniform space is a set with a uniform structure. (Def: http://en.m.wikipedia.org/wiki/Uniform_space)

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Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...
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Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all ...
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When is a uniform space complete

From Wikipedia: a uniform space is called complete if every Cauchy filter converges. I was wondering if the following three are equivalent in a uniform space: every Cauchy filter converges, ...
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Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
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A uniformly continuous function between totally bounded uniform spaces

Let $X$ and $Y$ is a uniform spaces. Let $f$ is a uniformly continuous surjective function $X\rightarrow Y$. Conjecture: If $X$ is totally bounded then $Y$ is also totally bounded.
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Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
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Showing a uniformity is complete.

I've seen in various textbooks and notes that if $X$ is paracompact, then the collection of all the neighborhoods of the diagonal is a uniformity. I am trying to show that this uniformity is complete ...
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What are interesting properties of totally bounded uniform spaces?

I work on reloids, a generalization of uniform spaces. As such I am interested about properties of totally bounded uniform spaces. My question: What are interesting properties of totally bounded ...
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TVS: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
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How can we construct *Fine uniformities*?

Given a uniformizable (w.r.t. entourage uniformity) space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. A uniform ...
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$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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Spaces with the property: Uniformly continuous equals continuous

I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ...
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non-hausdorff completion of a uniform space.

Let $(X,\mathcal U)$ be a Hausdorff uniform space. Can $(X,\mathcal U)$ have a non-hausdorff completion?
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Separation axioms in uniform spaces

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ ...
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Dieudonné complete and topologically complete are equivalent for every space $X$.

How can we show that: For every topological space $X$ the following conditions are equivalent: A space $X$ is topologically complete if $X$ is homeomorphic to a closed subspace of a product ...
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Is every $T_4$ topological space divisible?

A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the ...
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non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta X^2\subseteq\...
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Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
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The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
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closure of bounded set in uniform space

I have stumbled upon the following statement but I could not see why it is true. Let $(X,V)$ be a uniform space and $A$ bounded in it. Bounded means that for each $W\in V $ there is a finite set $F\...
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precompactness and boundedness in uniform space

Consider a uniform space $(X,\mathscr{U})$. For an entourage $U\in\mathscr{U}$, one says that a set $M$ is small of order $U$, if $M\times M\subseteq U$. $P\subset X$ is precompact if for every $U\in\...