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Definition df-retr 34209
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 34208 . 2 class Retr
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 22395 . . 3 class Top
5 vr . . . . . . . . 9 setvar 𝑟
65cv 1541 . . . . . . . 8 class 𝑟
7 vs . . . . . . . . 9 setvar 𝑠
87cv 1541 . . . . . . . 8 class 𝑠
96, 8ccom 5681 . . . . . . 7 class (𝑟𝑠)
10 cid 5574 . . . . . . . 8 class I
112cv 1541 . . . . . . . . 9 class 𝑗
1211cuni 4909 . . . . . . . 8 class 𝑗
1310, 12cres 5679 . . . . . . 7 class ( I ↾ 𝑗)
14 chtpy 24483 . . . . . . . 8 class Htpy
1511, 11, 14co 7409 . . . . . . 7 class (𝑗 Htpy 𝑗)
169, 13, 15co 7409 . . . . . 6 class ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗))
17 c0 4323 . . . . . 6 class
1816, 17wne 2941 . . . . 5 wff ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
193cv 1541 . . . . . 6 class 𝑘
20 ccn 22728 . . . . . 6 class Cn
2119, 11, 20co 7409 . . . . 5 class (𝑘 Cn 𝑗)
2218, 7, 21wrex 3071 . . . 4 wff 𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
2311, 19, 20co 7409 . . . 4 class (𝑗 Cn 𝑘)
2422, 5, 23crab 3433 . . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅}
252, 3, 4, 4, 24cmpo 7411 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
261, 25wceq 1542 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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