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Constructive Urysohn's Universal Metric Space

Davorin Lešnik
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
A construction of the Urysohn's universal metric space is given in the context of constructive theory of metric spaces.  ...  The space is universal in the sense that every separable metric space isometrically embeds into it.  ...  Urysohn's original idea was to construct a countable metric space of which U is the completion. We adopt the same approach here.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2008.12.015">doi:10.1016/j.entcs.2008.12.015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tpxmoijwavf4hesu5tuqo222ra">fatcat:tpxmoijwavf4hesu5tuqo222ra</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190305131336/https://core.ac.uk/download/pdf/82690887.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2a/d9/2ad91dfe5601dc30024a74ea5b16e9d844e52c68.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2008.12.015"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> elsevier.com </button> </a>

Finite Ball Intersection Property of the Urysohn Universal Space [article]

Asuman Guven Aksoy, Zair Ibragimov
<span title="2014-02-17">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space.  ...  Here we study various convexity properties of the Urysohn universal space and show that it has a finite ball intersection property. We also note that Urysohn universal space is not hyperconvex.  ...  Urysohn's construction Urysohn's original construction of U is published in full details in [19] . For the sake of completeness and accessibility we shall briefly go through the construction.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.7381v3">arXiv:1309.7381v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g4ktmyrxdjbcrbs6msbnvthr6u">fatcat:g4ktmyrxdjbcrbs6msbnvthr6u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200901105652/https://arxiv.org/pdf/1309.7381v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/14/db/14dbc8ef99fb714357845bb117525f141172f54d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.7381v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Urysohn universal space, its development and Hausdorff's approach

Miroslav Hušek
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hp7o5f6pmrg57k2m46s4syavf4" style="color: black;">Topology and its Applications</a> </i> &nbsp;
Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov.  ...  We shall not deal with some abstract constructions implying existence of Urysohn universal space like Jónsson classes or Fraïssé limits.  ...  Very probably, after reading the Urysohn's result he started to construct a universal space by his own.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.topol.2008.03.020">doi:10.1016/j.topol.2008.03.020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7vybwl3hrngcxmo64mvzaoyft4">fatcat:7vybwl3hrngcxmo64mvzaoyft4</a> </span>
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Page 487 of Mathematical Reviews Vol. 45, Issue 2 [page]

<span title="">1973</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
From the author’s introduction: “As one of his many accomplishments, Urysohn constructed a universal sepa- rable metric space U; that is, a separable metric space U such that any separable metric space  ...  Most of the theorems included stress metrization conditions for wM-spaces. D. R. Traylor (Houston, Tex.) Joiner, Charles On Urysohn’s universal separable metric space. Fund.  ... 
<span class="external-identifiers"> </span>
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Metrizability of multiset topological spaces

Karishma Shravan, Binod Chandra Tripathy
<span title="2021-01-20">2021</span> <i title="Universitatea Transilvania Brasov"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xpz5frrb6bgkxhovevtbbuksiu" style="color: black;">SERIES III - MATEMATICS, INFORMATICS, PHYSICS</a> </i> &nbsp;
Metrizable spaces are those topological spaces which are homeomorphic to a metric space.  ...  So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space.  ...  Remark 4 . 4 The empty mset and the universal mset in a metric space is always open. Lemma 1 . 1 Let (M, d) be a multiset metric space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.31926/but.mif.2020.13.62.2.24">doi:10.31926/but.mif.2020.13.62.2.24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4or4ujswlza55ka6bprdat466e">fatcat:4or4ujswlza55ka6bprdat466e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210926120013/http://webbut.unitbv.ro/bulletin/Series%20III/2020/BULETIN%20I/24.%20Shravan,%20Tripathy.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/20/57/205752303b159f61f62ffd6369e72751e2a67d87.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.31926/but.mif.2020.13.62.2.24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 1137 of Mathematical Reviews Vol. 16, Issue 11 [page]

<span title="">1955</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Metric spaces are now introduced for the first time; they are handled as special cases of pseudo-metric spaces.  ...  (Wallman’s construction is sketched in an exercise.)  ... 
<span class="external-identifiers"> </span>
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Essential Topology (by M. D. Crossley)

Ariel Blanco
<span title="">2006</span> <i title="Maynooth University"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3xa6g4wddjf2hnratcsok4c4qy" style="color: black;">Irish Mathematical Society Bulletin</a> </i> &nbsp;
Then it continues with metric spaces in Chapter 2. Several topological concepts are first discussed in this setting, although reference to metrics is avoided in the proofs whenever possible.  ...  The chapter starts with Urysohn's Lemma, Urysohn's metrization and Tietze's extension theorems.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.33232/bims.0057.99.100">doi:10.33232/bims.0057.99.100</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wcazennef5e7nkrepec25v5pgq">fatcat:wcazennef5e7nkrepec25v5pgq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20211016093421/https://www.irishmathsoc.org/bull57/BR5701.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bd/09/bd0907385aace08d60fca4183a886c3ff2e72489.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.33232/bims.0057.99.100"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Lipschitz functions on topometric spaces [article]

Itaï Ben Yaacov
<span title="2013-01-29">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used  ...  We also recover a compact topometric space X from the lattice of continuous 1-Lipschitz functions on X, in analogy with the recovery of a compact topological space X from the structure of (real or complex  ...  For example, existence results such as Urysohn's Lemma and Tietze's Extension Theorem are tied with normality, discussed in Section 1, while the Stone-Čech compactification (defined in terms of a universal  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1010.1600v2">arXiv:1010.1600v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/as5u6hkr75eltihdnqlf3jdoru">fatcat:as5u6hkr75eltihdnqlf3jdoru</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1010.1600/1010.1600.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bd/8d/bd8d8193f7f82fb6432e3647d3ca72e7ebbde05b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1010.1600v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Random Graph [chapter]

Peter J. Cameron
<span title="">2013</span> <i title="Springer New York"> The Mathematics of Paul Erdős II </i> &nbsp;
Urysohn's theorem A Polish space is a metric space which is complete (Cauchy sequences converge) and separable (there is a countable dense set).  ...  Instead, use the class of finite rational metric spaces (those with all distances rational). This is a Fraïssé class, whose Fraïssé limit is a countable universal homogeneous rational metric space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-1-4614-7254-4_22">doi:10.1007/978-1-4614-7254-4_22</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cayt2bi5bbcopopzvbh3a3yqai">fatcat:cayt2bi5bbcopopzvbh3a3yqai</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20160101131138/http://www-circa.mcs.st-and.ac.uk/~pjc/talks/spm14/pjc_rg2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d8/cc/d8ccb255292352ff22eb0258350bde0af3c7ff7c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-1-4614-7254-4_22"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The Random Graph [chapter]

Peter J. Cameron
<span title="">1997</span> <i title="Springer Berlin Heidelberg"> Algorithms and Combinatorics </i> &nbsp;
Urysohn's theorem A Polish space is a metric space which is complete (Cauchy sequences converge) and separable (there is a countable dense set).  ...  Instead, use the class of finite rational metric spaces (those with all distances rational). This is a Fraïssé class, whose Fraïssé limit is a countable universal homogeneous rational metric space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-60406-5_32">doi:10.1007/978-3-642-60406-5_32</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e7fxqhokovblfna2hrhvscsb3a">fatcat:e7fxqhokovblfna2hrhvscsb3a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20160101131138/http://www-circa.mcs.st-and.ac.uk/~pjc/talks/spm14/pjc_rg2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d8/cc/d8ccb255292352ff22eb0258350bde0af3c7ff7c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-60406-5_32"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Book Review: A treatise on set topology

Hing Tong
<span title="1948-11-01">1948</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
These two volumes on the theory of analytic functions are based on lectures given by the author at the University of Florence.  ...  Kolmogorov's theory of resolution spaces, chain resolution, Urysohn's lemma on normality, and Urysohn's extension theorem are treated.  ...  Convergence in metric spaces is treated; here analyses are made of compactness, completeness and topological completeness.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1948-09103-0">doi:10.1090/s0002-9904-1948-09103-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xaylak5ow5ewddh5yt2vauauam">fatcat:xaylak5ow5ewddh5yt2vauauam</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180724203810/http://www.ams.org/journals/bull/1948-54-11/S0002-9904-1948-09103-0/S0002-9904-1948-09103-0.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5a/29/5a2913e23e3fd267ae85e9750abebb059a3cdf5e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9904-1948-09103-0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Metric Baumgartner theorems and universality

Stefan Geschke, Menachem Kojman
<span title="">2007</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2kl46uxkrbdd7fdlgbwvbn6g4i" style="color: black;">Mathematical Research Letters</a> </i> &nbsp;
It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space U the following holds: (  ...  finite dimensional Banach space X, there is a unique universal element up to almost-isometry in the class of subspaces of X of size ℵ 1  ...  We thank the referee for pointing out that Fact 2.2 holds for separable metric spaces, not just for perfect Polish spaces.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/mrl.2007.v14.n2.a5">doi:10.4310/mrl.2007.v14.n2.a5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bzasjmbatncqvfkmqpp66icir4">fatcat:bzasjmbatncqvfkmqpp66icir4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180719211810/http://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2007/0014/0002/MRL-2007-0014-0002-a005.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6d/4f/6d4f06a2d9a02b01b22f234ab1f8c3f970775378.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/mrl.2007.v14.n2.a5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Lipschitz functions on topometric spaces

Itaï Ben Yaacov
<span title="">2013</span> <i title="Journal of Logic and Analysis"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3fseukthdbbrncufqe3im27wk4" style="color: black;">Journal of Logic and Analysis</a> </i> &nbsp;
We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used  ...  We also recover a compact topometric space X from the lattice of continuous 1-Lipschitz functions on X, in analogy with the recovery of a compact topological space X from the structure of (real or complex  ...  The identity (1) follows from Urysohn's Lemma for normal topometric spaces and the fact that a compact topometric space is normal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4115/jla.2013.5.8">doi:10.4115/jla.2013.5.8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/su7iafvr3bf4fk5y5phksdfq7e">fatcat:su7iafvr3bf4fk5y5phksdfq7e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20130409220627/http://hal.archives-ouvertes.fr/docs/00/78/23/74/PDF/LipTM.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d2/e5/d2e5b7311413cf25fcb436031de2b74625aac7f9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4115/jla.2013.5.8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Strong law of large numbers with concave moments [article]

Anders Karlsson, Nicolas Monod
<span title="2008-03-12">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
There is indeed a wealth of such metrics; recall that even Z admits an invariant metric whose completion is Urysohn's universal polish space [9] (by Theorem 4 in [2] ).  ...  Wild proper metrics can be constructed by means of weighted infinite generating sets. (iii) One can relax the concavity assumption is various ways.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0803.1856v1">arXiv:0803.1856v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cnilt2xj3rfzhpllithx76bt44">fatcat:cnilt2xj3rfzhpllithx76bt44</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0803.1856/0803.1856.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0803.1856v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Random and Universal Metric Spaces [chapter]

Anatoly M. Vershik
<span title="">2004</span> <i title="Springer Netherlands"> Dynamics and Randomness II </i> &nbsp;
A natural construction of a wide class of measures on the cone R is given and for these we show that with probability one a random Polish space is again the Urysohn space.  ...  This means that Urysohn space is generic in the set of all Polish spaces. Then we consider metric spaces with measures (metric triples) and define a complete invariant: its -matrix distribution.  ...  Thus we can construct a "random" metric space as the result of completion of the random metric on the natural numbers.  ... 
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