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• Homotopy equivalent topological spaces
  ...''homotopy equivalent'' to {{M|Y}}, or {{M|X}} and {{M|Y}} have the same ''homotopy type'', written {{M|X\simeq Y}}, if{{rITTMJML}}: ...ntinuous maps]] from {{M|X}} and {{M|f\simeq g}} denotes the relation of [[homotopy of maps]] - that is in this case [[freely homotopic]]
  3 KB (596 words) - 21:13, 24 April 2017

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• Homeomorphism
  ...a bijective continuous map, say {{M|f:X\rightarrow Y}}, the following are equivalent{{rITTMJML}}: If {{Top.|X|J}} and {{Top.|Y|K}} are [[topological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITTMJML}}:
  5 KB (731 words) - 22:58, 22 February 2017
• Site projects:Patrolling topology/Task list
  * [[Continuity definitions are equivalent]] * [[Homotopy class]]
  4 KB (404 words) - 21:36, 30 September 2016
• Notes:Types of retractions
  ...ace]] {{M|A}} of {{Top.|X|J}} is a ''deformation retract'' if there is a [[homotopy]], {{M|H:X\times I\rightarrow X}} - called a ''deformation'' such that: ...{M|A}} of {{Top.|X|J}} is a ''strong deformation retract'' if there is a [[homotopy]], {{M|H:X\times I\rightarrow X}} - called a ''deformation'' such that:
  6 KB (1,008 words) - 11:56, 2 June 2016
• Invariant of an equivalence relation
  ...(subject)|Linear Algebra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from * [[Homotopy invariance of path concatenation]]
  3 KB (478 words) - 18:58, 9 November 2016
• A continuous map induces a homomorphism on fundamental groups
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]] and let {{M|p\i ...{-1}([f]) }} in the notation developed next, giving us a set of all things equivalent to {{M|f}} and for any of these {{M|\varphi_\ast}} must yield the same resu
  8 KB (1,475 words) - 07:35, 14 December 2016
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
  # Prove that {{M|T^2}} and {{M|X}} are not [[homotopy equivalent spaces]] Observe that both {{M|X}} and {{M|T^2}} are [[path-connected topological spaces]]. As a result we will write {{M|\pi_1(X)}} or {{M|\pi_1(T^2)}} for their {
  10 KB (1,664 words) - 12:43, 1 March 2017
• Unique lifting property
  ...>{{rITTMJML}}^-<ref group="Note">Lee defines covering maps and spaces a little differently. He requires that for {{link|evenly covered|topology}} Recall that a [[logical implication]] is [[logically equivalent]] to the [[contrapositive]], that is
  13 KB (2,510 words) - 16:23, 2 March 2017
• Simply connected topological space
  ===Examples of simply connected spaces=== ...thbb{S}^{n-1} }} are [[homotopy equivalent topological spaces]] and then [[homotopy invariance of the fundamental group]] tells us {{M|\pi_1(\mathbb{R}^n-\{0\}
  4 KB (601 words) - 16:10, 24 April 2017
• Homotopy equivalent topological space
  #REDIRECT [[Homotopy equivalent topological spaces]] {{Definition|Topology|Algebraic Topology|Homotopy Theory}}
  111 B (11 words) - 20:03, 24 April 2017
• Homotopy equivalent topological spaces
  ...''homotopy equivalent'' to {{M|Y}}, or {{M|X}} and {{M|Y}} have the same ''homotopy type'', written {{M|X\simeq Y}}, if{{rITTMJML}}: ...ntinuous maps]] from {{M|X}} and {{M|f\simeq g}} denotes the relation of [[homotopy of maps]] - that is in this case [[freely homotopic]]
  3 KB (596 words) - 21:13, 24 April 2017
• Homotopy equivalence
  #REDIRECT [[Homotopy equivalent topological spaces]] {{Definition|Algebraic Topology|Topology|Homotopy Theory}}
  111 B (11 words) - 15:20, 15 December 2017