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Loop (topology) - Revision history
2024-03-28T12:31:36Z
Revision history for this page on the wiki
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http://www.maths.kisogo.com/index.php?title=Loop_(topology)&diff=3555&oldid=prev
Alec: /* See also */ Adding loop concatenation
2016-11-01T20:32:59Z
<p><span dir="auto"><span class="autocomment">See also: </span> Adding loop concatenation</span></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 20:32, 1 November 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The ''constant loop based at {{M|x_0\in X}}'' is the loop: {{M|\ell:[0,1]\rightarrow X}} given by {{M|\ell:t\mapsto x_0}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The ''constant loop based at {{M|x_0\in X}}'' is the loop: {{M|\ell:[0,1]\rightarrow X}} given by {{M|\ell:t\mapsto x_0}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Loop concatenation]] - creating a new loop, {{M|\ell_1*\ell_2}} from loops with the same basepoint, {{M|\ell_1}} and {{M|\ell_2}}.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Path]] and [[loop]]. These are always related no matter the context.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Path]] and [[loop]]. These are always related no matter the context.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>** [[Path (topology)]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>** [[Path (topology)]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Constant loop based at a point]] ({{AKA}}: [[constant loop]])</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Constant loop based at a point]] ({{AKA}}: [[constant loop]])</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[First homotopy group]], {{M|\pi_1(X,x_0)}} - a [[group]] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[First homotopy group]], {{M|\pi_1(X,x_0)}} - a [[group]] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Notes==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Notes==</div></td></tr>
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Alec
http://www.maths.kisogo.com/index.php?title=Loop_(topology)&diff=3318&oldid=prev
Alec: Created page with "{{Stub page|grade=A|msg=Created as a stub. Be sure to create disambiguation page loop.<br/> Tasks: # Don't be lazy, give full definition}} : '''Note: ''' see loop for..."
2016-10-15T00:23:02Z
<p>Created page with "{{Stub page|grade=A|msg=Created as a stub. Be sure to create disambiguation page <a href="/index.php?title=Loop" title="Loop">loop</a>.<br/> Tasks: # Don't be lazy, give full definition}} : '''Note: ''' see <a href="/index.php?title=Loop" title="Loop">loop</a> for..."</p>
<p><b>New page</b></p><div>{{Stub page|grade=A|msg=Created as a stub. Be sure to create disambiguation page [[loop]].<br/><br />
Tasks:<br />
# Don't be lazy, give full definition}}<br />
: '''Note: ''' see [[loop]] for other uses of the term.<br />
__TOC__<br />
==Definition==<br />
Let {{M|p:[0,1]\rightarrow X}} be a {{link|path|topology}} exactly as is defined on that page, then{{rITTMJML}}:<br />
* {{M|p}} is a ''loop'' if<ref group="Note">See also: [[Definitions and iff]]</ref>:<br />
** {{M|1=p(0)=p(1)}}, or in words: the initial point equals the terminal point of the path. <br />
We call {{M|1=p(0)=p(1)=x_0}} the ''base point'' of the loop. <br />
<br />
The ''constant loop based at {{M|x_0\in X}}'' is the loop: {{M|\ell:[0,1]\rightarrow X}} given by {{M|\ell:t\mapsto x_0}}<br />
==See also==<br />
* [[Path]] and [[loop]]. These are always related no matter the context.<br />
** [[Path (topology)]]<br />
* [[Constant loop based at a point]] ({{AKA}}: [[constant loop]])<br />
* [[First homotopy group]], {{M|\pi_1(X,x_0)}} - a [[group]] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}}<br />
==Notes==<br />
<references group="Note"/><br />
==References==<br />
<references/><br />
{{Definition|Topology|Metric Space}}</div>
Alec