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        Questions tagged [uniform-spaces]

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        5answers
        303 views

        Open mapping theorem for complete non-metrizable spaces?

        The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
        2
        votes
        1answer
        135 views

        Does each $\omega$-narrow topological group have countable discrete cellularity?

        A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable. A family $\mathcal F$ of subsets of a topological space ...
        2
        votes
        1answer
        82 views

        Uniformly Converging Metrization of Uniform Structure

        This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question. Let $X$ be a set with a uniform structure ...
        28
        votes
        3answers
        709 views

        What is the structure preserved by strong equivalence of metrics?

        Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
        4
        votes
        1answer
        120 views

        What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

        It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric What can say about $2^X= \{A\...
        3
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        0answers
        83 views

        Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

        Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...
        2
        votes
        1answer
        163 views

        Quotient of compact metrizable space in Hausdorff space

        Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
        7
        votes
        1answer
        414 views

        Totally bounded spaces and axiom of choice

        Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...
        4
        votes
        1answer
        110 views

        When is the unitary dual of a lscs group uniformizable?

        Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}...
        0
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        1answer
        70 views

        Topology generated by complete and incomplete uniformities [closed]

        Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?
        4
        votes
        1answer
        136 views

        Convergent net in a quasi-uniform space which is not Cauchy

        The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...
        7
        votes
        0answers
        165 views

        Results that are easier in a metric space

        Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces? In particular, I'm ...
        5
        votes
        0answers
        129 views

        Has the Roelcke completion of a topological group any reasonable algebraic structure?

        It is well-known that each topological group $G$ carries (at least) four natural uniformities: the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...
        2
        votes
        1answer
        80 views

        Cartesian powers of uniform spaces

        In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped ...
        2
        votes
        1answer
        100 views

        Construct a specific base for Fine uniformities in the diagonal(Entourages) case

        For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. To construct Fine uniformities, Let ...

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