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Definition df-retr 35223
Description: Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽𝐾, 𝑆:𝐾𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅𝑆 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 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
Distinct variable group:   𝑗,𝑘,𝑟,𝑠

Detailed syntax breakdown of Definition df-retr
StepHypRef Expression
1 cretr 35222 . 2 class Retr
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 22899 . . 3 class Top
5 vr . . . . . . . . 9 setvar 𝑟
65cv 1539 . . . . . . . 8 class 𝑟
7 vs . . . . . . . . 9 setvar 𝑠
87cv 1539 . . . . . . . 8 class 𝑠
96, 8ccom 5689 . . . . . . 7 class (𝑟𝑠)
10 cid 5577 . . . . . . . 8 class I
112cv 1539 . . . . . . . . 9 class 𝑗
1211cuni 4907 . . . . . . . 8 class 𝑗
1310, 12cres 5687 . . . . . . 7 class ( I ↾ 𝑗)
14 chtpy 24999 . . . . . . . 8 class Htpy
1511, 11, 14co 7431 . . . . . . 7 class (𝑗 Htpy 𝑗)
169, 13, 15co 7431 . . . . . 6 class ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗))
17 c0 4333 . . . . . 6 class
1816, 17wne 2940 . . . . 5 wff ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
193cv 1539 . . . . . 6 class 𝑘
20 ccn 23232 . . . . . 6 class Cn
2119, 11, 20co 7431 . . . . 5 class (𝑘 Cn 𝑗)
2218, 7, 21wrex 3070 . . . 4 wff 𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
2311, 19, 20co 7431 . . . 4 class (𝑗 Cn 𝑘)
2422, 5, 23crab 3436 . . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅}
252, 3, 4, 4, 24cmpo 7433 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
261, 25wceq 1540 1 wff Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
Colors of variables: wff setvar class
This definition is referenced by: (None)
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