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Definition df-retr 34204
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 34203 . 2 class Retr
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 22394 . . 3 class Top
5 vr . . . . . . . . 9 setvar 𝑟
65cv 1540 . . . . . . . 8 class 𝑟
7 vs . . . . . . . . 9 setvar 𝑠
87cv 1540 . . . . . . . 8 class 𝑠
96, 8ccom 5680 . . . . . . 7 class (𝑟𝑠)
10 cid 5573 . . . . . . . 8 class I
112cv 1540 . . . . . . . . 9 class 𝑗
1211cuni 4908 . . . . . . . 8 class 𝑗
1310, 12cres 5678 . . . . . . 7 class ( I ↾ 𝑗)
14 chtpy 24482 . . . . . . . 8 class Htpy
1511, 11, 14co 7408 . . . . . . 7 class (𝑗 Htpy 𝑗)
169, 13, 15co 7408 . . . . . 6 class ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗))
17 c0 4322 . . . . . 6 class
1816, 17wne 2940 . . . . 5 wff ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
193cv 1540 . . . . . 6 class 𝑘
20 ccn 22727 . . . . . 6 class Cn
2119, 11, 20co 7408 . . . . 5 class (𝑘 Cn 𝑗)
2218, 7, 21wrex 3070 . . . 4 wff 𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
2311, 19, 20co 7408 . . . 4 class (𝑗 Cn 𝑘)
2422, 5, 23crab 3432 . . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅}
252, 3, 4, 4, 24cmpo 7410 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
261, 25wceq 1541 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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