<html> <head> <title>Lviv seminar on Topological Algebra archivta_2008-2009</title> <Base > <meta http-equiv="??????????-???" content="text/html; charset=iso-8859-1"> <META name="keywords" CONTENT="Oleg Gutik Igor Guran Olexander Ravsky"> <!-- *************** DHTML Outline (begin) ***************** --> <STYLE TYPE='text/css'> <!-- /*Define elements with children with the class='outlineParentItem', and all elements without children with class='outlineItem'*/ li.oItem { color: #000000; cursor: text; } ; li.oParent { color: #000088; cursor: hand; } ; ul ul { display: none; } ; // --> </STYLE> <SCRIPT LANGUAGE='Javascript'> <!-- // Returns the closest parent tag with tagName containing // the src tag. If no such tag is found - null is returned. function checkParent( src, tagName ) { while ( src != null ) { if (src.tagName == tagName) return src; src = src.parentElement; } return null; } // Returns the first tag with tagName contained by // the src tag. If no such tag is found - null is returned. function checkContent( src, tagName ) { var pos = src.sourceIndex ; while ( src.contains( document.all[++pos] ) ) if ( document.all[pos].tagName == tagName ) return document.all[pos] ; return null ; } // Handle onClick event in the outline box function outlineAction() { var src = event.srcElement ; var item = checkParent( src, "LI" ) ; if ( parent != null ) { var content = checkContent( item, "UL" ) ; if ( content != null ) if ( content.style.display == "" ) content.style.display = "block" ; else content.style.display = "" ; } event.cancelBubble = true; } // --> </SCRIPT> </head> <body background="backgr_3.gif" bgcolor="#FFCC66" text="#660000" vlink="#990000" alink="#990066" link="#990000"leftmargin="10" rightmargin="0" topmargin="1" > <Table width="95%" border=0 align="center"> <tr> <td > <img src="emb_1.gif"> </td> <td align="center"> <b><font size="+2"><a href=http://www.franko.lviv.ua/faculty/mechmat/Departments/Topology/seminarta.html> Scientific Seminar</a></font> <font size="+1"><br>at Geometry and Topology Department of <br> Ivan Franko National University of Lviv <br> <a href="seminarta.html"><img src="left.gif"></a> </b><i>Archive for</i><b> <font size="+2">(2008/2009)</font> </b><i>academic year</i></b></font> <a href="seminararchiv_ta09_10.html"><img src="right.gif"></a> </td> <td align="center"> <p> <b> <font size="+1"> <hr> <hr> Topological <br> Algebra <hr> <hr> </font></b> </p> </tr> </table> <Table width="92%" border=0> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> November 27, 2008 </font><td valign="top" width="15%" > <b><font size="+2"> O. Ravsky </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Tkachenko-Tomita Group Example Construction <UL type="square"><LI class='oItem'> <font size="+1"> A topological group G is called a Tkachenko-Tomita if G is countably compact, abelian, torsion-free and contains no no-trivial convergent sequence. The first example of a Tkachenko-Tomita group was constructed by M. Tkachenko under the Continuum Hypothesis. Later, the Continuum Hypothesis was weakened to the Martin Axiom for &sigma;-centered posets by A. Tomita, for countable posets by P. Koszmider, A. Tomita and S. Watson, and finally to the existence continuum many incomparable selective ultrafilters by R. Madariaga-Garcia and A. Tomita. Yet, no ZFC-example of a Tkachenko-Tomita group is known. The aim of our meeting is to understand the construction by P. Koszmider, A. Tomita and S. Watson. </LI></UL></font></ul></tr> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> December 4, 2008,<br> December 11, 2008<br> </font><td valign="top" width="15%" > <b><font size="+2"> O. Ravsky </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On algebraically Baire topological groups <UL type="square"><LI class='oItem'> <font size="+1"> A topological group $G$ is algebraically Baire if $CN\not=G$ for each countable subset $C$ of $G$ and each nowhere dense subset $N$ of $G$. A topological group $G$ is locally algebraically Baire if $int (CN)=\emptyset$ for each countable subset $C$ of $G$ and each nowhere dense subset $N$ of $G$. Every Baire topological group is locally algebraically Baire and every locally algebraically Baire topological group is algebraically Baire. These facts suggest to consider the following questions. Question 1. Is every locally algebraically Baire topological group a Baire group? Question 2. Is every algebraically Baire topological group a locally algebraically Baire group? It seems that we have obtained the following partial answers to these questions. Answer 1. Every not precompact locally algebraically Baire topological group is a Baire group. Answer 2. Every $\omega$-precompact algebraically Baire topological group is a locally algebraically Baire group. </LI></UL></font></ul></tr> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> February 12, 2009 </font><td valign="top" width="15%" > <b><font size="+2"> <a href="guran.html"> I. Guran</a>, <br> <a href="Gutik_mine.html">O. Gutik</a> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Divertissement <UL type="square"><LI class='oItem'> <font size="+1"> Some open problems on the theory of topological semigroups were possed </LI></UL></font></ul></tr> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> February 19, 2009,<br> February 26, 2009,<br> March 5, 2009 </font><td valign="top" width="15%" > <b><font size="+2"> <a href="guran.html"> I. Guran</a> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Embedding of a cancellative topological semigroup into a topological group <UL type="square"><LI class='oItem'> <font size="+1"> The problem of an embedding of topological semigroups into topological groups will be discuss. </LI></UL></font></ul></tr> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> March 12, 2009<br> </font><td valign="top" width="15%" > <b><font size="+2"> O. Ravsky </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Characterizing meager paratopological groups <UL type="square"><LI class='oItem'> <font size="+1"> We are going to finish the proof of the next<br> <b>Theorem.</b> A topological group $G$ is meager if and only if there is a nowhere dense subset $A\subset G$ and a countable subset $C\subset G$ such that $C\cdot A=G$. <br> At the seminar we shall consider the case when the group is precompact. </LI></UL></font></ul></tr> <tr> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> </table> <br> <a href="https://plus.google.com/u/0/+OlegGutik"><img src="Rgplus-Gutik.jpg"></a> <a href="https://plus.google.com/u/0/+OlGutikPallady"><img src="Rgplus-Gutik.jpg"></a> <a href="https://plus.google.com/u/0/+TopologicalAlgebraSeminar"><img src="Rgplus-Gutik.jpg"></a> <br> <!-- Start of StatCounter Code for Default Guide --> <script type="text/javascript"> var sc_project=9174828; var sc_invisible=0; var sc_security="d274098c"; var scJsHost = (("https:" == document.location.protocol) ? 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