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Definition df-retr 32573
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 32572 . 2 class Retr
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 21501 . . 3 class Top
5 vr . . . . . . . . 9 setvar 𝑟
65cv 1537 . . . . . . . 8 class 𝑟
7 vs . . . . . . . . 9 setvar 𝑠
87cv 1537 . . . . . . . 8 class 𝑠
96, 8ccom 5527 . . . . . . 7 class (𝑟𝑠)
10 cid 5427 . . . . . . . 8 class I
112cv 1537 . . . . . . . . 9 class 𝑗
1211cuni 4803 . . . . . . . 8 class 𝑗
1310, 12cres 5525 . . . . . . 7 class ( I ↾ 𝑗)
14 chtpy 23575 . . . . . . . 8 class Htpy
1511, 11, 14co 7139 . . . . . . 7 class (𝑗 Htpy 𝑗)
169, 13, 15co 7139 . . . . . 6 class ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗))
17 c0 4246 . . . . . 6 class
1816, 17wne 2990 . . . . 5 wff ((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
193cv 1537 . . . . . 6 class 𝑘
20 ccn 21832 . . . . . 6 class Cn
2119, 11, 20co 7139 . . . . 5 class (𝑘 Cn 𝑗)
2218, 7, 21wrex 3110 . . . 4 wff 𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅
2311, 19, 20co 7139 . . . 4 class (𝑗 Cn 𝑘)
2422, 5, 23crab 3113 . . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅}
252, 3, 4, 4, 24cmpo 7141 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})
261, 25wceq 1538 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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