<front xsi:noNamespaceSchemaLocation="C:/programs/XMLTOXHTML/NLM/jats-journalpublishing1.xsd" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><journal-meta><journal-id journal-id-type="publisher-id">IJMMS</journal-id><journal-title-group><journal-title>International Journal of Mathematics and Mathematical Sciences</journal-title></journal-title-group><issn pub-type="epub">1687-0425</issn><issn pub-type="ppub">0161-1712</issn><publisher><publisher-name>Hindawi Publishing Corporation</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">408261</article-id><article-id pub-id-type="doi">10.1155/S016117128000049X</article-id><title-group><article-title>Peano compactifications and property <mml:math alttext="$S$" id="E1"><mml:mi>S</mml:mi></mml:math> metric spaces</article-title></title-group><contrib-group><contrib contrib-type="author" id="U84535038"><name><surname>Dickman</surname><given-names>R. F.</given-names><suffix>Jr.</suffix></name><xref ref-type="aff" rid="I1" /></contrib></contrib-group><aff id="I1"><addr-line>Department of Mathematics</addr-line><addr-line>Virginia Polytechnic Institute and State University</addr-line><addr-line>Blacksburg, Virginia 24061</addr-line><country>USA</country><ext-link ext-link-type="domain-name">vt.edu</ext-link></aff><pub-date pub-type="publication-year"><year>1980</year></pub-date><volume>3</volume><issue>4</issue><fpage>695</fpage><lpage>700</lpage><history><date date-type="received"><day>17</day><month>01</month><year>1980</year></date></history><permissions><copyright-year>1980</copyright-year><copyright-holder>Copyright © 1980 Hindawi Publishing Corporation</copyright-holder><license license-type="open-access"><license-p>This is an open access article distributed under the <ext-link xlink:href="http://creativecommons.org/licenses/by/3.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p></license></permissions><abstract><p>Let <mml:math alttext="$\left( {X,d} \right)$" id="E2"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> denote a locally connected, connected separable metric space. We say the <mml:math alttext="$X$" id="E3"><mml:mi>X</mml:mi></mml:math> is <mml:math alttext="$S$" id="E4"><mml:mi>S</mml:mi></mml:math>-metrizable provided there is a topologically equivalent metric <mml:math alttext="$\rho $" id="E5"><mml:mi>ρ</mml:mi></mml:math> on <mml:math alttext="$X$" id="E6"><mml:mi>X</mml:mi></mml:math> such that <mml:math alttext="$\left( {X,\rho } \right)$" id="E7"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> has Property <mml:math alttext="$S$" id="E8"><mml:mi>S</mml:mi></mml:math>, i.e. for any <mml:math alttext="$\varepsilon  &gt; 0$" id="E9"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, <mml:math alttext="$X$" id="E10"><mml:mi>X</mml:mi></mml:math> is the union of finitely many connected sets of <mml:math alttext="$\rho $" id="E11"><mml:mi>ρ</mml:mi></mml:math>-diameter less than <mml:math alttext="$\varepsilon $" id="E12"><mml:mi>ϵ</mml:mi></mml:math>. It is well-known that <mml:math alttext="$S$" id="E13"><mml:mi>S</mml:mi></mml:math>-metrizable spaces are locally connected and that if <mml:math alttext="$\rho $" id="E14"><mml:mi>ρ</mml:mi></mml:math> is a Property <mml:math alttext="$S$" id="E15"><mml:mi>S</mml:mi></mml:math> metric for <mml:math alttext="$X$" id="E16"><mml:mi>X</mml:mi></mml:math>, then the usual metric completion <mml:math alttext="$\left( {\tilde X,\tilde \rho } \right)$" id="E17"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> of <mml:math alttext="$\left( {X,\rho } \right)$" id="E18"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is a compact, locally connected, connected metric space, i.e. <mml:math alttext="$\left( {\tilde X,\tilde \rho } \right)$" id="E19"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mi>ρ</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is a Peano compactification of <mml:math alttext="$\left( {X,\rho } \right)$" id="E20"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>. There are easily constructed examples of locally connected connected metric spaces which fail to be <mml:math alttext="$S$" id="E21"><mml:mi>S</mml:mi></mml:math>-metrizable, however the author does not know of a non-<mml:math alttext="$S$" id="E22"><mml:mi>S</mml:mi></mml:math>-metrizable space <mml:math alttext="$\left( {X,d} \right)$" id="E23"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> which has a Peano compactification. In this paper we conjecture that: If <mml:math alttext="$\left( {P,\rho } \right)$" id="E24"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> a Peano compactification of <mml:math alttext="$\left( {X,\rho \left| X \right.} \right)$" id="E25"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo>|</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math alttext="$X$" id="E26"><mml:mi>X</mml:mi></mml:math> must be <mml:math alttext="$S$" id="E27"><mml:mi>S</mml:mi></mml:math>-metrizable. Several (new) necessary and sufficient for a space to be <mml:math alttext="$S$" id="E28"><mml:mi>S</mml:mi></mml:math>-metrizable are given, together with an example of non-<mml:math alttext="$S$" id="E29"><mml:mi>S</mml:mi></mml:math>-metrizable space which fails to have a Peano compactification.</p></abstract><kwd-group><kwd>property <italic>S</italic> metrics</kwd><kwd>Peano spaces</kwd><kwd>compactifications</kwd></kwd-group><counts><ref-count count="8" /><page-count count="6" /></counts></article-meta></front>
