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Definition df-retr 24676
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 24675 . 2  class Retr
2 vj . . 3  set  j
3 vk . . 3  set  k
4 ctop 16874 . . 3  class  Top
5 vr . . . . . . . . 9  set  r
65cv 1648 . . . . . . . 8  class  r
7 vs . . . . . . . . 9  set  s
87cv 1648 . . . . . . . 8  class  s
96, 8ccom 4815 . . . . . . 7  class  ( r  o.  s )
10 cid 4427 . . . . . . . 8  class  _I
112cv 1648 . . . . . . . . 9  class  j
1211cuni 3950 . . . . . . . 8  class  U. j
1310, 12cres 4813 . . . . . . 7  class  (  _I  |`  U. j )
14 chtpy 18856 . . . . . . . 8  class Htpy
1511, 11, 14co 6013 . . . . . . 7  class  ( j Htpy  j )
169, 13, 15co 6013 . . . . . 6  class  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )
17 c0 3564 . . . . . 6  class  (/)
1816, 17wne 2543 . . . . 5  wff  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
193cv 1648 . . . . . 6  class  k
20 ccn 17203 . . . . . 6  class  Cn
2119, 11, 20co 6013 . . . . 5  class  ( k  Cn  j )
2218, 7, 21wrex 2643 . . . 4  wff  E. s  e.  ( k  Cn  j
) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
2311, 19, 20co 6013 . . . 4  class  ( j  Cn  k )
2422, 5, 23crab 2646 . . 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 6015 . 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 1649 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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