http://helmut.knaust.info/mediawiki/index.php?action=history&feed=atom&title=CRN_12109%3A_HW_5 2026-05-16T04:45:01Z Revision history for this page on the wiki MediaWiki 1.19.1 http://helmut.knaust.info/mediawiki/index.php?title=CRN_12109:_HW_5&diff=864&oldid=prev 2013-10-24T20:58:28Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 20:58, 24 October 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 15:</td> <td colspan="2" class="diff-lineno">Line 15:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Show that a set is LP-compact if and only if it is closed and bounded.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Show that a set is LP-compact if and only if it is closed and bounded.</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Hint: Show first that every bounded infinite set has a limit point.</ins></div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=CRN_12109:_HW_5&diff=856&oldid=prev 2013-10-18T13:50:15Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 13:50, 18 October 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 12:</td> <td colspan="2" class="diff-lineno">Line 12:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem 25.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $<del class="diffchange diffchange-inline">A</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 25.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $<ins class="diffchange diffchange-inline">X</ins>$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Show that a set is LP-compact if and only if it is closed and bounded.</div></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"><div>Show that a set is LP-compact if and only if it is closed and bounded.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=CRN_12109:_HW_5&diff=855&oldid=prev 2013-10-15T22:47:16Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 22:47, 15 October 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 12:</td> <td colspan="2" class="diff-lineno">Line 12:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem 25.''' A set $X$ is called ''<del class="diffchange diffchange-inline">limit point </del>compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $A$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 25.''' A set $X$ is called ''<ins class="diffchange diffchange-inline">LP-</ins>compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $A$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Show that a set is <del class="diffchange diffchange-inline">limit point </del>compact if and only if it is closed and bounded.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Show that a set is <ins class="diffchange diffchange-inline">LP-</ins>compact if and only if it is closed and bounded.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=CRN_12109:_HW_5&diff=853&oldid=prev 2013-10-15T22:44:38Z

<p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 22:44, 15 October 2013</td> </tr><tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">'''Problem 21.''' A set $X$ is called ''limit point compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $A$.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Show that a set is limit point compact if and only if it is closed and bounded</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 21</ins>.<ins class="diffchange diffchange-inline">''' Exercise 3.2.2 (a-d)</ins></div></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'>&#160;</td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">21</del>.''' Exercise 2.<del class="diffchange diffchange-inline">7</del>.<del class="diffchange diffchange-inline">6</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">22.''' Given a set $X$ of real numbers, let $L$ be the set of all limit points of $X$. Show that $L$ is closed.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 23.''' Show: If $X$ is both open and closed, then $X={\mathbb R}$ or $X=\emptyset$.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 24</ins>.''' Exercise <ins class="diffchange diffchange-inline">3.</ins>2.<ins class="diffchange diffchange-inline">14</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 25.''' A set $X$ is called ''limit point compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $A$.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Show that a set is limit point compact if and only if it is closed and bounded</ins>.</div></td></tr> </table>

HelmutKnaust http://helmut.knaust.info/mediawiki/index.php?title=CRN_12109:_HW_5&diff=852&oldid=prev 2013-10-15T22:36:43Z

<p>Created page with &quot;&#039;&#039;&#039;Problem 21.&#039;&#039;&#039; A set $X$ is called &#039;&#039;limit point compact&#039;&#039;, if every infinite subset $A$ of $X$ has a limit point belonging to $A$. Show that a set is limit point compact ...&quot;</p> <p><b>New page</b></p><div>'''Problem 21.''' A set $X$ is called ''limit point compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $A$.<br /> <br /> Show that a set is limit point compact if and only if it is closed and bounded.<br /> <br /> <br /> '''Problem 21.''' Exercise 2.7.6</div>

HelmutKnaust