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Definition df-retr 23749
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 23748 . 2  class Retr
2 vj . . 3  set  j
3 vk . . 3  set  k
4 ctop 16631 . . 3  class  Top
5 vr . . . . . . . . 9  set  r
65cv 1622 . . . . . . . 8  class  r
7 vs . . . . . . . . 9  set  s
87cv 1622 . . . . . . . 8  class  s
96, 8ccom 4693 . . . . . . 7  class  ( r  o.  s )
10 cid 4304 . . . . . . . 8  class  _I
112cv 1622 . . . . . . . . 9  class  j
1211cuni 3827 . . . . . . . 8  class  U. j
1310, 12cres 4691 . . . . . . 7  class  (  _I  |`  U. j )
14 chtpy 18465 . . . . . . . 8  class Htpy
1511, 11, 14co 5858 . . . . . . 7  class  ( j Htpy  j )
169, 13, 15co 5858 . . . . . 6  class  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )
17 c0 3455 . . . . . 6  class  (/)
1816, 17wne 2446 . . . . 5  wff  ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
193cv 1622 . . . . . 6  class  k
20 ccn 16954 . . . . . 6  class  Cn
2119, 11, 20co 5858 . . . . 5  class  ( k  Cn  j )
2218, 7, 21wrex 2544 . . . 4  wff  E. s  e.  ( k  Cn  j
) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j
) )  =/=  (/)
2311, 19, 20co 5858 . . . 4  class  ( j  Cn  k )
2422, 5, 23crab 2547 . . 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 5860 . 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 1623 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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