Mathematical Researches

fa گروه‌وار بنیادین و توپولوژی ویسکر Fundamental groupoid and Whisker topology ریاضی Mat مقاله استخراج شده از پایان نامه Research Paper این مقاله به معرفی توپولوژی ویسکر بر گروه‌وار بنیادین و ارتباط این توپولوژی با توپولوژی ویسکر بر گروه‌ بنیادین می‌پردازد. همچنین شرایطی را ارائه خواهد داد که از گروه‌وار بنیادین یک گروه‌وار توپولوژیکی بدست آید. Introduction Algebraic structures on topological spaces can’t distinguish all the non-homeomorphic spaces. Recently, by equipping some of these algebraic structures with topology, one can separate the non-homeomorphic spaces with the same algebraic structures. Particularly, the fundamental group is equipped by the various types of topologies which some of them make it a topological group and some of them don't&lrm;. Virk&lrm; &lrm;and&lrm; &lrm;Zastro&lrm;w has&lrm; &lrm;studied&lrm; &lrm;the&lrm; &lrm;generalizations&lrm; &lrm;of&lrm; &lrm;the&lrm; &lrm;existing&lrm;&lrm;&lrm; &lrm;topologies&lrm; &lrm;on&lrm; &lrm;the&lrm; &lrm;fundamental&lrm;&lrm;group&lrm; &lrm;to&lrm; &lrm;the&lrm; &lrm;universal&lrm; &lrm;path&lrm; &lrm;space&lrm; with a proper comparison. If we remove the condition α(0) = <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>x</m:r></m:e><m:sub><m:r>0</m:r></m:sub></m:ssub></m:omath>  from the definition of the universal path space,  we obtain an object that have been discussed in mathematical literatures under the name  “ fundamental groupoid”. Indeed, fundamental groupoid denoted by <m:omath><m:r>π</m:r><m:r>X</m:r></m:omath> , is the category of homotopy classes of paths in <m:omath><m:r>X</m:r></m:omath>  as the morphisms and has the set <m:omath><m:r>X</m:r></m:omath>  as the objects set. For any <m:omath><m:r>x</m:r></m:omath> ,<m:omath><m:r> </m:r><m:r>y</m:r></m:omath>  <m:omath><m:r>∈</m:r><m:r>X</m:r></m:omath> , the set <m:omath><m:r>π</m:r><m:r>X</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></m:e></m:d></m:omath>  is the set of homotopy classes of paths in <m:omath><m:r>X</m:r></m:omath>  from <m:omath><m:r>x</m:r></m:omath>  to <m:omath><m:r>y</m:r></m:omath> . We can consider the object group at <m:omath><m:r>x</m:r></m:omath> , <m:omath><m:r>π</m:r><m:r>X</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>x</m:r></m:e></m:d></m:omath> , as the well-known fundamental group <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>π</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>X</m:r><m:r>,</m:r><m:r>x</m:r></m:e></m:d><m:r>.</m:r></m:omath>  &lrm; &lrm; In order to answer the question how these topologies can be generalized on the fundamental groupoids, the authors has introduced the Lasso topology on the fundamental groupoid of a locally path connected space in which makes it a topological groupoid. A topological groupoid is a groupoid <m:omath><m:r>G</m:r></m:omath>  together with topologies on <m:omath><m:r>G</m:r></m:omath>  and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>G</m:r></m:e><m:sub><m:r>0</m:r></m:sub></m:ssub></m:omath>  such that the structure maps are continuous&lrm;.  R&lrm;. &lrm;Brown and G&lrm;. &lrm;Danesh-Naruie were the first and only ones to take this step&lrm;. &lrm;They have defined a topology on a quotient of the fundamental groupoid such that it became a topological groupoid when the given space <m:omath><m:r>X</m:r></m:omath>  is locally path connected and semilocally simply connected. Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>X</m:r></m:omath>  in which it’s basis is known and by some assumptions, we can consider it as a generalization of the whisker topology on the fundamental group. Material and methods &lrm;&lrm; For a given topological space <m:omath><m:r>(</m:r><m:r>X</m:r><m:r>,</m:r><m:r>τ</m:r><m:r>)</m:r></m:omath> , let <m:omath><m:r>[</m:r><m:r>α</m:r><m:r>]∈</m:r><m:r>π</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>X</m:r><m:r>(</m:r><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r><m:r>)</m:r></m:omath> &lrm;, where <m:omath><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r><m:r>∈</m:r><m:r>X</m:r></m:omath> . &lrm;If <m:omath><m:r>V</m:r></m:omath> , &lrm;<m:omath><m:r> </m:r><m:r>W</m:r></m:omath>  are open neighborhoods of <m:omath><m:r>x</m:r></m:omath> &lrm;, y, respectively&lrm;, one can define <m:omath><m:r>N</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>α</m:r></m:e></m:d><m:r>,</m:r><m:r>V</m:r><m:r>,</m:r><m:r>W</m:r></m:e></m:d><m:r>:=</m:r><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="|"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>β</m:r></m:e></m:d><m:r>∈</m:r><m:r>πX</m:r></m:e></m:d><m:r> </m:r><m:r>β</m:r><m:r>≃</m:r><m:r>γ</m:r><m:r>*</m:r><m:r>α</m:r><m:r>*</m:r><m:r>λ</m:r><m:r>, </m:r><m:r>γ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>&sube;</m:r><m:r>V</m:r><m:r>, </m:r><m:r>λ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>&sube;</m:r><m:r>W</m:r><m:r>}</m:r></m:omath> , where <m:omath><m:r>γ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>1</m:r></m:e></m:d><m:r>=</m:r><m:r>α</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>0</m:r></m:e></m:d><m:r>=</m:r><m:r>x</m:r></m:omath>  and <m:omath><m:r>α</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>1</m:r></m:e></m:d><m:r>=</m:r><m:r>λ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>0</m:r></m:e></m:d><m:r>=</m:r><m:r>y</m:r><m:r>.</m:r></m:omath> Theorem: &lrm;The family&lrm; &lrm;<m:omath><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>N</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>α</m:r></m:e></m:d><m:r>,</m:r><m:r>V</m:r><m:r>,</m:r><m:r>W</m:r></m:e></m:d><m:r>;</m:r><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>α</m:r></m:e></m:d><m:r>∈</m:r><m:r>πX</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></m:e></m:d><m:r>,</m:r><m:r>x</m:r><m:r>∈</m:r><m:r>V</m:r><m:r>∈</m:r><m:r>τ</m:r><m:r>,</m:r><m:r>y</m:r><m:r>∈</m:r><m:r>W</m:r><m:r>∈</m:r><m:r>τ</m:r></m:e></m:d></m:omath> &lrm;forms a basis for a topology on fundamental groupoid&lrm;.  &lrm;The topology that is generated by this basis&lrm;, &lrm;is called Whisker topology. Conclusion Based on the results that we presented in this paper: If <m:omath><m:r>X</m:r></m:omath>  is a small loop transfer space; The multiplication map <m:omath><m:r>m</m:r><m:r>: </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>π</m:r></m:e><m:sup><m:r>w</m:r><m:r>h</m:r></m:sup></m:ssup><m:r>X</m:r><m:r> × </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>π</m:r></m:e><m:sup><m:r>w</m:r><m:r>h</m:r></m:sup></m:ssup><m:r>X</m:r><m:r> → </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>π</m:r></m:e><m:sup><m:r>w</m:r><m:r>h</m:r></m:sup></m:ssup><m:r>X</m:r><m:r> </m:r></m:omath> is continuous. The fundamental groupoid with the Whisker topology is a topological groupoid. The inherited topology from fundamental groupoid <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>π</m:r></m:e><m:sup><m:r>w</m:r><m:r>h</m:r></m:sup></m:ssup><m:r>X</m:r></m:omath>  on the object group <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>π</m:r></m:e><m:sup><m:r>w</m:r><m:r>h</m:r></m:sup></m:ssup><m:r>X</m:r><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></m:omath>  equals to the Whisker topology on <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>π</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>X</m:r><m:r>,</m:r><m:r>x</m:r></m:e></m:d><m:r>.</m:r></m:omath>   گروه‌وار بنیادین, توپولوژی ویسکر, گروه‌وار توپولوژیکی Fundamental groupoid, Whisker topology, Topological groupoid 55 71 http://mmr.khu.ac.ir/browse.php?a_code=A-10-744-1&slc_lang=fa&sid=1 Ali Pakdaman علی پاکدامن a.pakdaman@gu.ac.ir 10031947532846005988 10031947532846005988 Yes دانشگاه گلستان Freshteh Shahini فرشته شاهینی freshte.shahini@gmail.com 10031947532846005989 10031947532846005989 No دانشگاه گلستان