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There is some disagreement about which definition is the "standard" definition of local contractibility; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to
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if it is locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In
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is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).
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Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by
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is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally
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standard
Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.
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topological properties. Care should always be taken regarding the definitions when interpreting results about these properties.
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Illustration of some contractible and non-contractible spaces. Spaces A, B, and C are contractible; spaces D, E, and F are not.
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to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
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is contractible (since it is a cone), but not locally contractible or even locally simply connected.
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by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose
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is a standard example of a space which is contractible, but not intuitively so.
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onto a point. (However, there exist contractible spaces which do not
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is contractible (i.e. the identity map is null-homotopic).
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279:of contractible neighborhoods, then we say that
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326:, C.R.. Acad. Sci. Paris 199 (1934), 110-112).
62:A contractible space is precisely one with the
414:contractible, but in general not contractible.
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360:of any finite dimension are not contractible.
429:are all trivial, but it is not contractible.
102:is homotopy equivalent to a one-point space.
442: – Mathematical topological manifold
324:Sur les rétractes absolus indécomposables
82:of a contractible space are all trivial.
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16:Can be continuously shrunk to a point
66:of a point. It follows that all the
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89:the following are all equivalent:
85:For a non-empty topological space
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50:is null-homotopic, i.e. if it is
543:Properties of topological spaces
421:is obtained by "closing up" the
115:deformation retract to a point.)
228:locally contractible at a point
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299:≥ 0. In particular, they are
285:strongly locally contractible
118:For any path-connected space
78:is a homotopy invariant, the
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334:Examples and counterexamples
195:Every contractible space is
70:of a contractible space are
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367:in an infinite-dimensional
343:is contractible, as is any
256:such that the inclusion of
218:Locally contractible spaces
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503:Cambridge University Press
388:is contractible, but not
301:locally simply connected
244:there is a neighborhood
423:topologist's sine curve
80:reduced homology groups
305:locally path connected
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347:on a Euclidean space.
275:If every point has a
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379:house with two rooms
266:locally contractible
222:A topological space
109:deformation retracts
295:-connected for all
260:is nulhomotopic in
498:Algebraic Topology
352:Whitehead manifold
157:is null-homotopic.
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461:Munkres, James R.
309:locally connected
184:from the cone of
76:singular homology
33:topological space
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467:(2nd ed.).
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397:Hawaiian earring
354:is contractible.
201:simply connected
176:is contractible
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538:Homotopy theory
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372:is contractible
341:Euclidean space
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322:in their paper
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180:there exists a
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68:homotopy groups
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395:The cone on a
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264:. A space is
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197:path connected
178:if and only if
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141:For any space
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138:are homotopic.
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44:identity map
40:contractible
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440:Fake 4-ball
390:collapsible
365:unit sphere
345:star domain
165:on a space
29:mathematics
527:Categories
447:References
289:comb space
277:local base
208:-connected
182:retraction
145:, any map
58:Properties
404:manifolds
386:Dunce hat
270:Hatcher's
52:homotopic
533:Topology
495:(2002).
465:Topology
463:(2000).
434:See also
210:for all
113:strongly
412:locally
358:Spheres
72:trivial
42:if the
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316:Borsuk
307:, and
214:≥ 0.
507:ISBN
473:ISBN
417:The
410:are
406:and
402:All
384:The
377:The
363:The
350:The
339:Any
318:and
199:and
163:cone
161:The
31:, a
283:is
248:of
240:of
226:is
188:to
46:on
38:is
27:In
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128:g
126:,
124:f
120:Y
106:X
100:X
94:X
87:X
48:X
36:X
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