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instiki_ncatlab.org.atom.xml (502820B)
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1 <feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en">
2 <title>nLab</title>
3 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/HomePage"/>
4 <link rel="self" href="https://ncatlab.org/nlab/atom_with_content"/>
5 <updated>2021-07-02T09:22:44Z</updated>
6 <id>tag:ncatlab.org,2008-11-28:nLab</id>
7 <subtitle>An Instiki Wiki</subtitle>
8 <generator uri="http://golem.ph.utexas.edu/instiki/show/HomePage" version="0.19.7(MML+)">Instiki</generator>
9 <entry>
10 <title type="html">Urs Frauenfelder</title>
11 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/Urs+Frauenfelder"/>
12 <updated>2021-07-02T09:22:44Z</updated>
13 <published>2021-07-02T09:22:46Z</published>
14 <id>tag:ncatlab.org,2021-07-02:nLab,Urs+Frauenfelder</id>
15 <author>
16 <name>Urs Schreiber</name>
17 </author>
18 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Urs+Frauenfelder">
19 <div xmlns="http://www.w3.org/1999/xhtml">
20 <ul>
21 <li>
22 <p><a href='https://www.uni-augsburg.de/en/fakultaet/mntf/math/prof/geom/frauenfelder/'>Institute page</a></p>
23 </li>
24
25 <li>
26 <p><a href='https://www.genealogy.math.ndsu.nodak.edu/id.php?id=120240'>MathematicsGenealogy page</a></p>
27 </li>
28 </ul>
29
30 <h2 id='selected_writings'>Selected writings</h2>
31
32 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cyclic loop spaces</a>:</p>
33
34 <ul>
35 <li><a class='existingWikiWord' href='/nlab/show/Urs+Frauenfelder'>Urs Frauenfelder</a>, <em>Dihedral homology and the moon</em>, J. Fixed Point Theory Appl. <strong>14</strong> (2013) 55–69 (<a href='https://arxiv.org/abs/1204.4549'>arXiv:1204.4549</a>, <a href='https://doi.org/10.1007/s11784-013-0146-z'>doi:10.1007/s11784-013-0146-z</a>)</li>
36 </ul>
37
38 <p><div class='property'> category: <a class='category_link' href='/nlab/list/people'>people</a></div></p> </div>
39 </content>
40 </entry>
41 <entry>
42 <title type="html">cyclic loop space</title>
43 <link rel="alternate" type="application/xhtml+xml" href="https://ncatlab.org/nlab/show/cyclic+loop+space"/>
44 <updated>2021-07-02T09:20:18Z</updated>
45 <published>2017-02-14T09:56:17Z</published>
46 <id>tag:ncatlab.org,2017-02-14:nLab,cyclic+loop+space</id>
47 <author>
48 <name>Urs Schreiber</name>
49 </author>
50 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/cyclic+loop+space">
51 <div xmlns="http://www.w3.org/1999/xhtml">
52 <div class='rightHandSide'>
53 <div class='toc clickDown' tabindex='0'>
54 <h3 id='context'>Context</h3>
55
56 <h4 id='topology'>Topology</h4>
57
58 <div class='hide'>
59 <p><strong><a class='existingWikiWord' href='/nlab/show/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/point-free+topology'>point-free topology</a>)</p>
60
61 <p>see also <em><a class='existingWikiWord' href='/nlab/show/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></em></p>
62
63 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology'>Introduction</a></p>
64
65 <p><strong>Basic concepts</strong></p>
66
67 <ul>
68 <li>
69 <p><a class='existingWikiWord' href='/nlab/show/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/neighborhood'>neighbourhood</a></p>
70 </li>
71
72 <li>
73 <p><a class='existingWikiWord' href='/nlab/show/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/locale'>locale</a></p>
74 </li>
75
76 <li>
77 <p><a class='existingWikiWord' href='/nlab/show/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/neighborhood+base'>neighbourhood base</a></p>
78 </li>
79
80 <li>
81 <p><a class='existingWikiWord' href='/nlab/show/finer+topology'>finer/coarser topology</a></p>
82 </li>
83
84 <li>
85 <p><a class='existingWikiWord' href='/nlab/show/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/boundary'>boundary</a></p>
86 </li>
87
88 <li>
89 <p><a class='existingWikiWord' href='/nlab/show/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/sober+topological+space'>sobriety</a></p>
90 </li>
91
92 <li>
93 <p><a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/homeomorphism'>homeomorphism</a></p>
94 </li>
95
96 <li>
97 <p><a class='existingWikiWord' href='/nlab/show/uniformly+continuous+map'>uniformly continuous function</a></p>
98 </li>
99
100 <li>
101 <p><a class='existingWikiWord' href='/nlab/show/embedding+of+topological+spaces'>embedding</a></p>
102 </li>
103
104 <li>
105 <p><a class='existingWikiWord' href='/nlab/show/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/closed+map'>closed map</a></p>
106 </li>
107
108 <li>
109 <p><a class='existingWikiWord' href='/nlab/show/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/filter'>filter</a></p>
110 </li>
111
112 <li>
113 <p><a class='existingWikiWord' href='/nlab/show/convergence'>convergence</a></p>
114 </li>
115
116 <li>
117 <p><a class='existingWikiWord' href='/nlab/show/category'>category</a> <a class='existingWikiWord' href='/nlab/show/Top'>Top</a></p>
118
119 <ul>
120 <li><a class='existingWikiWord' href='/nlab/show/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li>
121 </ul>
122 </li>
123 </ul>
124
125 <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p>
126
127 <ul>
128 <li>
129 <p><a class='existingWikiWord' href='/nlab/show/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/weak+topology'>final topology</a></p>
130 </li>
131
132 <li>
133 <p><a class='existingWikiWord' href='/nlab/show/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/quotient+space'>quotient space</a>,</p>
134 </li>
135
136 <li>
137 <p>fiber space, <a class='existingWikiWord' href='/nlab/show/space+attachment'>space attachment</a></p>
138 </li>
139
140 <li>
141 <p><a class='existingWikiWord' href='/nlab/show/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/disjoint+union+topological+space'>disjoint union space</a></p>
142 </li>
143
144 <li>
145 <p><a class='existingWikiWord' href='/nlab/show/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/cocylinder'>mapping cocylinder</a></p>
146 </li>
147
148 <li>
149 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
150 </li>
151
152 <li>
153 <p><a class='existingWikiWord' href='/nlab/show/mapping+telescope'>mapping telescope</a></p>
154 </li>
155
156 <li>
157 <p><a class='existingWikiWord' href='/nlab/show/colimits+of+normal+spaces'>colimits of normal spaces</a></p>
158 </li>
159 </ul>
160
161 <p><strong><a class='existingWikiWord' href='/nlab/show/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p>
162
163 <ul>
164 <li>
165 <p><a class='existingWikiWord' href='/nlab/show/nice+topological+space'>nice topological space</a></p>
166 </li>
167
168 <li>
169 <p><a class='existingWikiWord' href='/nlab/show/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/metrisable+topological+space'>metrisable space</a></p>
170 </li>
171
172 <li>
173 <p><a class='existingWikiWord' href='/nlab/show/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/normal+space'>normal space</a></p>
174 </li>
175
176 <li>
177 <p><a class='existingWikiWord' href='/nlab/show/sober+topological+space'>sober space</a></p>
178 </li>
179
180 <li>
181 <p><a class='existingWikiWord' href='/nlab/show/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/proper+map'>proper map</a></p>
182
183 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/strongly+compact+topological+space'>strongly compact</a></p>
184 </li>
185
186 <li>
187 <p><a class='existingWikiWord' href='/nlab/show/compactly+generated+topological+space'>compactly generated space</a></p>
188 </li>
189
190 <li>
191 <p><a class='existingWikiWord' href='/nlab/show/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/first-countable+space'>first-countable space</a></p>
192 </li>
193
194 <li>
195 <p><a class='existingWikiWord' href='/nlab/show/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/locally+contractible+space'>locally contractible space</a></p>
196 </li>
197
198 <li>
199 <p><a class='existingWikiWord' href='/nlab/show/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/locally+connected+topological+space'>locally connected space</a></p>
200 </li>
201
202 <li>
203 <p><a class='existingWikiWord' href='/nlab/show/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p>
204 </li>
205
206 <li>
207 <p><a class='existingWikiWord' href='/nlab/show/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/CW+complex'>CW-complex</a></p>
208 </li>
209
210 <li>
211 <p><a class='existingWikiWord' href='/nlab/show/pointed+topological+space'>pointed space</a></p>
212 </li>
213
214 <li>
215 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/Hilbert+space'>Hilbert space</a></p>
216 </li>
217
218 <li>
219 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topological group</a></p>
220 </li>
221
222 <li>
223 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/topological+K-theory'>topological K-theory</a></p>
224 </li>
225
226 <li>
227 <p><a class='existingWikiWord' href='/nlab/show/topological+manifold'>topological manifold</a></p>
228 </li>
229 </ul>
230
231 <p><strong>Examples</strong></p>
232
233 <ul>
234 <li>
235 <p><a class='existingWikiWord' href='/nlab/show/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/point+space'>point space</a></p>
236 </li>
237
238 <li>
239 <p><a class='existingWikiWord' href='/nlab/show/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/codiscrete+space'>codiscrete space</a></p>
240 </li>
241
242 <li>
243 <p><a class='existingWikiWord' href='/nlab/show/Sierpinski+space'>Sierpinski space</a></p>
244 </li>
245
246 <li>
247 <p><a class='existingWikiWord' href='/nlab/show/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/Scott+topology'>Scott topology</a></p>
248 </li>
249
250 <li>
251 <p><a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclidean space</a></p>
252
253 <ul>
254 <li><a class='existingWikiWord' href='/nlab/show/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/plane'>plane</a></li>
255 </ul>
256 </li>
257
258 <li>
259 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/cone'>cone</a></p>
260 </li>
261
262 <li>
263 <p><a class='existingWikiWord' href='/nlab/show/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/ball'>ball</a></p>
264 </li>
265
266 <li>
267 <p><a class='existingWikiWord' href='/nlab/show/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/M%C3%B6bius+strip'>Moebius strip</a></p>
268 </li>
269
270 <li>
271 <p><a class='existingWikiWord' href='/nlab/show/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/polyhedron'>polyhedron</a></p>
272 </li>
273
274 <li>
275 <p><a class='existingWikiWord' href='/nlab/show/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/complex+projective+space'>complex</a>)</p>
276 </li>
277
278 <li>
279 <p><a class='existingWikiWord' href='/nlab/show/classifying+space'>classifying space</a></p>
280 </li>
281
282 <li>
283 <p><a class='existingWikiWord' href='/nlab/show/configuration+space+of+points'>configuration space</a></p>
284 </li>
285
286 <li>
287 <p><a class='existingWikiWord' href='/nlab/show/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/loop'>loop</a></p>
288 </li>
289
290 <li>
291 <p><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/topology+of+uniform+convergence'>topology of uniform convergence</a></p>
292
293 <ul>
294 <li><a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/path+space'>path space</a></li>
295 </ul>
296 </li>
297
298 <li>
299 <p><a class='existingWikiWord' href='/nlab/show/Zariski+topology'>Zariski topology</a></p>
300 </li>
301
302 <li>
303 <p><a class='existingWikiWord' href='/nlab/show/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/Mandelbrot+set'>Mandelbrot space</a></p>
304 </li>
305
306 <li>
307 <p><a class='existingWikiWord' href='/nlab/show/Peano+curve'>Peano curve</a></p>
308 </li>
309
310 <li>
311 <p><a class='existingWikiWord' href='/nlab/show/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/Sorgenfrey+line'>Sorgenfrey line</a></p>
312 </li>
313
314 <li>
315 <p><a class='existingWikiWord' href='/nlab/show/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/Dowker+space'>Dowker space</a></p>
316 </li>
317
318 <li>
319 <p><a class='existingWikiWord' href='/nlab/show/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/Hawaiian+earring+space'>Hawaiian earring space</a></p>
320 </li>
321 </ul>
322
323 <p><strong>Basic statements</strong></p>
324
325 <ul>
326 <li>
327 <p><a class='existingWikiWord' href='/nlab/show/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p>
328 </li>
329
330 <li>
331 <p><a class='existingWikiWord' href='/nlab/show/schemes+are+sober'>schemes are sober</a></p>
332 </li>
333
334 <li>
335 <p><a class='existingWikiWord' href='/nlab/show/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p>
336 </li>
337
338 <li>
339 <p><a class='existingWikiWord' href='/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p>
340 </li>
341
342 <li>
343 <p><a class='existingWikiWord' href='/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p>
344 </li>
345
346 <li>
347 <p><a class='existingWikiWord' href='/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p>
348 </li>
349
350 <li>
351 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net'>compact spaces equivalently have converging subnet of every net</a></p>
352
353 <ul>
354 <li>
355 <p><a class='existingWikiWord' href='/nlab/show/Lebesgue+number+lemma'>Lebesgue number lemma</a></p>
356 </li>
357
358 <li>
359 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p>
360 </li>
361
362 <li>
363 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net'>compact spaces equivalently have converging subnet of every net</a></p>
364 </li>
365
366 <li>
367 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p>
368 </li>
369 </ul>
370 </li>
371
372 <li>
373 <p><a class='existingWikiWord' href='/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p>
374 </li>
375
376 <li>
377 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p>
378 </li>
379
380 <li>
381 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p>
382 </li>
383
384 <li>
385 <p><a class='existingWikiWord' href='/nlab/show/closed+injections+are+embeddings'>closed injections are embeddings</a></p>
386 </li>
387
388 <li>
389 <p><a class='existingWikiWord' href='/nlab/show/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p>
390 </li>
391
392 <li>
393 <p><a class='existingWikiWord' href='/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p>
394 </li>
395
396 <li>
397 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p>
398 </li>
399
400 <li>
401 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p>
402 </li>
403
404 <li>
405 <p><a class='existingWikiWord' href='/nlab/show/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p>
406 </li>
407
408 <li>
409 <p><a class='existingWikiWord' href='/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p>
410 </li>
411 </ul>
412
413 <p><strong>Theorems</strong></p>
414
415 <ul>
416 <li>
417 <p><a class='existingWikiWord' href='/nlab/show/Urysohn%27s+lemma'>Urysohn's lemma</a></p>
418 </li>
419
420 <li>
421 <p><a class='existingWikiWord' href='/nlab/show/Tietze+extension+theorem'>Tietze extension theorem</a></p>
422 </li>
423
424 <li>
425 <p><a class='existingWikiWord' href='/nlab/show/Tychonoff+theorem'>Tychonoff theorem</a></p>
426 </li>
427
428 <li>
429 <p><a class='existingWikiWord' href='/nlab/show/tube+lemma'>tube lemma</a></p>
430 </li>
431
432 <li>
433 <p><a class='existingWikiWord' href='/nlab/show/Michael%27s+theorem'>Michael's theorem</a></p>
434 </li>
435
436 <li>
437 <p><a class='existingWikiWord' href='/nlab/show/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p>
438 </li>
439
440 <li>
441 <p><a class='existingWikiWord' href='/nlab/show/topological+invariance+of+dimension'>topological invariance of dimension</a></p>
442 </li>
443
444 <li>
445 <p><a class='existingWikiWord' href='/nlab/show/Jordan+curve+theorem'>Jordan curve theorem</a></p>
446 </li>
447 </ul>
448
449 <p><strong>Analysis Theorems</strong></p>
450
451 <ul>
452 <li>
453 <p><a class='existingWikiWord' href='/nlab/show/Heine-Borel+theorem'>Heine-Borel theorem</a></p>
454 </li>
455
456 <li>
457 <p><a class='existingWikiWord' href='/nlab/show/intermediate+value+theorem'>intermediate value theorem</a></p>
458 </li>
459
460 <li>
461 <p><a class='existingWikiWord' href='/nlab/show/extreme+value+theorem'>extreme value theorem</a></p>
462 </li>
463 </ul>
464
465 <p><strong><a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological homotopy theory</a></strong></p>
466
467 <ul>
468 <li>
469 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
470 </li>
471
472 <li>
473 <p><a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/deformation+retract'>deformation retract</a></p>
474 </li>
475
476 <li>
477 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/covering+space'>covering space</a></p>
478 </li>
479
480 <li>
481 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
482 </li>
483
484 <li>
485 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
486 </li>
487
488 <li>
489 <p><a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>weak homotopy equivalence</a></p>
490 </li>
491
492 <li>
493 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead's theorem</a></p>
494 </li>
495
496 <li>
497 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
498 </li>
499
500 <li>
501 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
502 </li>
503
504 <li>
505 <p><a class='existingWikiWord' href='/nlab/show/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/Hurewicz+cofibration'>Hurewicz cofibration</a></p>
506 </li>
507
508 <li>
509 <p><a class='existingWikiWord' href='/nlab/show/topological+cofiber+sequence'>cofiber sequence</a></p>
510 </li>
511
512 <li>
513 <p><a class='existingWikiWord' href='/nlab/show/Str%C3%B8m+model+structure'>Strøm model category</a></p>
514 </li>
515
516 <li>
517 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p>
518 </li>
519 </ul>
520 </div>
521
522 <h4 id='homotopy_theory'>Homotopy theory</h4>
523
524 <div class='hide'>
525 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a></strong></p>
526
527 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1%29-topos'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/directed+homotopy+theory'>directed</a>…</p>
528
529 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/localic+homotopy+theory'>localic</a>, …</p>
530
531 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong></p>
532
533 <p><strong>Introductions</strong></p>
534
535 <ul>
536 <li>
537 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p>
538 </li>
539
540 <li>
541 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p>
542 </li>
543
544 <li>
545 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p>
546 </li>
547 </ul>
548
549 <p><strong>Definitions</strong></p>
550
551 <ul>
552 <li>
553 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/higher+homotopy'>higher homotopy</a></p>
554 </li>
555
556 <li>
557 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p>
558 </li>
559
560 <li>
561 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/spherical+object'>spherical object and Pi(A)-algebra</a></p>
562 </li>
563
564 <li>
565 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p>
566
567 <ul>
568 <li>
569 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></p>
570
571 <ul>
572 <li>
573 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p>
574 </li>
575
576 <li>
577 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/cofibration+category'>cofibration category</a></p>
578 </li>
579
580 <li>
581 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Waldhausen category</a></p>
582 </li>
583 </ul>
584 </li>
585
586 <li>
587 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></p>
588
589 <ul>
590 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a></li>
591 </ul>
592 </li>
593 </ul>
594 </li>
595
596 <li>
597 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p>
598
599 <ul>
600 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li>
601 </ul>
602 </li>
603 </ul>
604
605 <p><strong>Paths and cylinders</strong></p>
606
607 <ul>
608 <li>
609 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a></p>
610
611 <ul>
612 <li>
613 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a></p>
614 </li>
615
616 <li>
617 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a></p>
618 </li>
619 </ul>
620 </li>
621
622 <li>
623 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p>
624
625 <ul>
626 <li>
627 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path object</a></p>
628 </li>
629
630 <li>
631 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p>
632 </li>
633
634 <li>
635 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle'>universal bundle</a></p>
636 </li>
637 </ul>
638 </li>
639
640 <li>
641 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a></p>
642
643 <ul>
644 <li>
645 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometric+homotopies'>homotopy localization</a></p>
646 </li>
647
648 <li>
649 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+object'>infinitesimal interval object</a></p>
650 </li>
651 </ul>
652 </li>
653 </ul>
654
655 <p><strong>Homotopy groups</strong></p>
656
657 <ul>
658 <li>
659 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p>
660
661 <ul>
662 <li>
663 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a></p>
664
665 <ul>
666 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+topos'>fundamental group of a topos</a></li>
667 </ul>
668 </li>
669
670 <li>
671 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p>
672 </li>
673
674 <li>
675 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p>
676 </li>
677
678 <li>
679 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p>
680 </li>
681 </ul>
682 </li>
683
684 <li>
685 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p>
686
687 <ul>
688 <li>
689 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a></p>
690
691 <ul>
692 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path groupoid</a></li>
693 </ul>
694 </li>
695
696 <li>
697 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p>
698 </li>
699
700 <li>
701 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p>
702 </li>
703 </ul>
704 </li>
705
706 <li>
707 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p>
708
709 <ul>
710 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental category</a></li>
711 </ul>
712 </li>
713 </ul>
714
715 <p><strong>Basic facts</strong></p>
716
717 <ul>
718 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li>
719 </ul>
720
721 <p><strong>Theorems</strong></p>
722
723 <ul>
724 <li>
725 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p>
726 </li>
727
728 <li>
729 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p>
730 </li>
731
732 <li>
733 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p>
734 </li>
735
736 <li>
737 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p>
738 </li>
739
740 <li>
741 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p>
742 </li>
743
744 <li>
745 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead's theorem</a></p>
746 </li>
747
748 <li>
749 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewicz theorem</a></p>
750 </li>
751
752 <li>
753 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois theory</a></p>
754 </li>
755
756 <li>
757 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p>
758 </li>
759 </ul>
760 </div>
761 </div>
762 </div>
763
764 <h1 id='contents'>Contents</h1>
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