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Definition df-retr 35164
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 35163 . 2 class Retr
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 22897 . . 3 class Top
5 vr . . . . . . . . 9 setvar 𝑟
65cv 1534 . . . . . . . 8 class 𝑟
7 vs . . . . . . . . 9 setvar 𝑠
87cv 1534 . . . . . . . 8 class 𝑠
96, 8ccom 5688 . . . . . . 7 class (𝑟𝑠)
10 cid 5576 . . . . . . . 8 class I
112cv 1534 . . . . . . . . 9 class 𝑗
1211cuni 4915 . . . . . . . 8 class 𝑗
1310, 12cres 5686 . . . . . . 7 class ( I ↾ 𝑗)
14 chtpy 24995 . . . . . . . 8 class Htpy
1511, 11, 14co 7426 . . . . . . 7 class (𝑗 Htpy 𝑗)
169, 13, 15co 7426 . . . . . 6 class ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗))
17 c0 4339 . . . . . 6 class
1816, 17wne 2936 . . . . 5 wff ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
193cv 1534 . . . . . 6 class 𝑘
20 ccn 23230 . . . . . 6 class Cn
2119, 11, 20co 7426 . . . . 5 class (𝑘 Cn 𝑗)
2218, 7, 21wrex 3066 . . . 4 wff 𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
2311, 19, 20co 7426 . . . 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 7428 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
261, 25wceq 1535 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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