Definition df-retr 32467

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:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 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)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})

Distinct variable group:   𝑗,𝑘,𝑟,𝑠

Detailed syntax breakdown of Definition df-retr

Step	Hyp	Ref	Expression
1		cretr 32466	. 2 class Retr
2		vj	. . 3 setvar 𝑗
3		vk	. . 3 setvar 𝑘
4		ctop 21503	. . 3 class Top
5		vr	. . . . . . . . 9 setvar 𝑟
6	5	cv 1536	. . . . . . . 8 class 𝑟
7		vs	. . . . . . . . 9 setvar 𝑠
8	7	cv 1536	. . . . . . . 8 class 𝑠
9	6, 8	ccom 5561	. . . . . . 7 class (𝑟 ∘ 𝑠)
10		cid 5461	. . . . . . . 8 class I
11	2	cv 1536	. . . . . . . . 9 class 𝑗
12	11	cuni 4840	. . . . . . . 8 class ∪ 𝑗
13	10, 12	cres 5559	. . . . . . 7 class ( I ↾ ∪ 𝑗)
14		chtpy 23573	. . . . . . . 8 class Htpy
15	11, 11, 14	co 7158	. . . . . . 7 class (𝑗 Htpy 𝑗)
16	9, 13, 15	co 7158	. . . . . 6 class ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗))
17		c0 4293	. . . . . 6 class ∅
18	16, 17	wne 3018	. . . . 5 wff ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
19	3	cv 1536	. . . . . 6 class 𝑘
20		ccn 21834	. . . . . 6 class Cn
21	19, 11, 20	co 7158	. . . . 5 class (𝑘 Cn 𝑗)
22	18, 7, 21	wrex 3141	. . . 4 wff ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
23	11, 19, 20	co 7158	. . . 4 class (𝑗 Cn 𝑘)
24	22, 5, 23	crab 3144	. . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}
25	2, 3, 4, 4, 24	cmpo 7160	. 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
26	1, 25	wceq 1537	1 wff Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})

This definition is referenced by: (None)