<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0370-3908</journal-id>
<journal-title><![CDATA[Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. acad. colomb. cienc. exact. fis. nat.]]></abbrev-journal-title>
<issn>0370-3908</issn>
<publisher>
<publisher-name><![CDATA[Academia Colombiana de Ciencias Exactas, Físicas y Naturales]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0370-39082017000400528</article-id>
<article-id pub-id-type="doi">10.18257/raccefyn.510</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Introducción de la topología de vecindades en los trabajos de Fréchet y Hausdorff]]></article-title>
<article-title xml:lang="en"><![CDATA[Introduction of the topology structure in Fréchet and Hausdorff works]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Arboleda]]></surname>
<given-names><![CDATA[Luis Carlos]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad del Valle  ]]></institution>
<addr-line><![CDATA[Cali ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2017</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2017</year>
</pub-date>
<volume>41</volume>
<numero>161</numero>
<fpage>528</fpage>
<lpage>537</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0370-39082017000400528&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0370-39082017000400528&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0370-39082017000400528&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En la primera parte, retomamos testimonios de Fréchet sobre la naturaleza de sus primeros trabajos (1904-1906) en los campos emergentes del Análisis funcional y Análisis general, en relación con su idea de introducir una estructura topológica en un espacio abstracto. En la segunda parte, destacamos la influencia que tuvo en esta idea, el punto de vista algebraico de la época de extender las nociones cantorianas a un espacio abstracto con una estructura de grupo finito. Fréchet supo aprovechar técnicas como el &#8220;modo de composición&#8221; entre los elementos del espacio, para axiomatizar operaciones y estructuras de la &#8220;clase L&#8221; con convergencia secuencial, la &#8220;clase V&#8221; con sistema de vecindades, la &#8220;clase E&#8221; con &#8220;écart&#8221; (métrica). Luego se aprovechan nuevos datos históricos para reafirmar la proximidad de las concepciones filosóficas subyacentes a estas investigaciones, con las ideas de Leibniz, específicamente en cuanto al método de &#8220;análisis de los principios&#8221;. En la tercera parte se estudia la contribución de Hausdorff de 1912 y 1914 al establecimiento de la axiomática de las vecindades para la topología de un espacio abstracto. Teniendo en cuenta las observaciones de Weyl y Bourbaki de que Hausdorff se inspiró para ello en Hilbert, se examina el sistema de axiomas para las vecindades del plano introducido por Hilbert en dos trabajos de 1902 consagrados al problema de la continuidad del espacio. Se exploran las conexiones del &#8220;espacio topológico&#8221; de Hausdorff basado en las vecindades, con las nociones de métrica, convergencia secuencial y vecindades propuestas años antes por Fréchet. Hausdorff insistió desde el comienzo que la topología del espacio separable tenía las características de generalidad y rigor formal que le permitían adaptarse a las aplicaciones mejor que otras. Se mostrará que todo ello era consistente con los ideales de simplicidad, unidad y economía de pensamiento que Hausdorff había adquirido en sus trabajos filosóficos tempranos.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In the first part, we remind Fréchet&#8217;s testimonies about the approach of his early work (1904-1906) in the emerging fields of Functional Analysis and General Analysis, in relation to his idea of introducing a topological structure in an abstract space. In the second part, we highlight the influence that had on this idea, the algebraic point of view of the time of extending the Cantorian notions to an abstract space with a finite group structure. Fréchet took advantage of techniques such as the &#8220;composition mode&#8221; between the elements of the space, to axiomatize operations and structures of the &#8220;class &#8221; with sequential convergence, the &#8220;class &#8221; with neighborhood system, the &#8220;class &#8221; with &#8220;écart&#8221; (metric). Then, new historical data are used to reaffirm the proximity of the philosophical conceptions underlying these investigations with the ideas of Leibniz, specifically with regard to the method of &#8220;analysis of principles&#8221;. The third part examines the contribution of Hausdorff of 1912 and 1914 to the establishment of the neighborhood axiomatics for the topology of an abstract space. Taking into account the observations of Weyl and Bourbaki that Hausdorff drew on Hilbert, we examine the system of axioms for the neighborhoods of the plane introduced by Hilbert in two 1902 papers devoted to the problem of the space continuity. As regards Fréchet&#8217;s influence in Hausdorff, we explore the connections of Hausdorff&#8217;s &#8220;topological space&#8221; based on neighborhoods with the notions of metric, sequential convergence and neighborhoods, proposed years earlier by Fréchet. From the beginning, Hausdorff argued that the topology of the separable space had the characteristics of generality and formal rigor that allowed it to adapt to applications better than others. It is shown that all this was consistent with the ideals of simplicity, unity and economy of thought Hausdorff had acquired in his early philosophical works.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[Historia de la topología]]></kwd>
<kwd lng="es"><![CDATA[Filosofía de la práctica matemática]]></kwd>
<kwd lng="es"><![CDATA[Sistemas de axiomas para vecindades]]></kwd>
<kwd lng="es"><![CDATA[Fréchet]]></kwd>
<kwd lng="es"><![CDATA[Hausdorff]]></kwd>
<kwd lng="en"><![CDATA[History of topology]]></kwd>
<kwd lng="en"><![CDATA[Philosophy of mathematical practice]]></kwd>
<kwd lng="en"><![CDATA[systems of axioms for neighborhoods]]></kwd>
<kwd lng="en"><![CDATA[Fréchet]]></kwd>
<kwd lng="en"><![CDATA[Hausdorff]]></kwd>
</kwd-group>
</article-meta>
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