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Definition df-retr 23764
Description: Define the set of retractions on two topological spaces. We say that  R is a retraction from  J to  K. or  R  e.  ( J Retr  K ) iff there is an  S such that  R : J --> K ,  S : K
--> J are continuous functions called the retraction and section respectively, and their composite  R  o.  S is homotopic to the identity map. If a retraction exists, we say  J is a retract of  K. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
df-retr  |- Retr  =  ( j  e.  Top , 
k  e.  Top  |->  { r  e.  ( j  Cn  k )  |  E. s  e.  ( k  Cn  j ) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j ) )  =/=  (/) } )
Distinct variable group:    j, k, r, s

Detailed syntax breakdown of Definition df-retr
StepHypRef Expression
1 cretr 23763 . 2  class Retr
2 vj . . 3  set  j
3 vk . . 3  set  k
4 ctop 16647 . . 3  class  Top
5 vr . . . . . . . . 9  set  r
65cv 1631 . . . . . . . 8  class  r
7 vs . . . . . . . . 9  set  s
87cv 1631 . . . . . . . 8  class  s
96, 8ccom 4709 . . . . . . 7  class  ( r  o.  s )
10 cid 4320 . . . . . . . 8  class  _I
112cv 1631 . . . . . . . . 9  class  j
1211cuni 3843 . . . . . . . 8  class  U. j
1310, 12cres 4707 . . . . . . 7  class  (  _I  |`  U. j )
14 chtpy 18481 . . . . . . . 8  class Htpy
1511, 11, 14co 5874 . . . . . . 7  class  ( j Htpy  j )
169, 13, 15co 5874 . . . . . 6  class  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )
17 c0 3468 . . . . . 6  class  (/)
1816, 17wne 2459 . . . . 5  wff  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
193cv 1631 . . . . . 6  class  k
20 ccn 16970 . . . . . 6  class  Cn
2119, 11, 20co 5874 . . . . 5  class  ( k  Cn  j )
2218, 7, 21wrex 2557 . . . 4  wff  E. s  e.  ( k  Cn  j
) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
2311, 19, 20co 5874 . . . 4  class  ( j  Cn  k )
2422, 5, 23crab 2560 . . 3  class  { r  e.  ( j  Cn  k )  |  E. s  e.  ( k  Cn  j ) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/) }
252, 3, 4, 4, 24cmpt2 5876 . 2  class  ( j  e.  Top ,  k  e.  Top  |->  { r  e.  ( j  Cn  k )  |  E. s  e.  ( k  Cn  j ) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/) } )
261, 25wceq 1632 1  wff Retr  =  ( j  e.  Top , 
k  e.  Top  |->  { r  e.  ( j  Cn  k )  |  E. s  e.  ( k  Cn  j ) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j ) )  =/=  (/) } )
Colors of variables: wff set class
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