Definition df-retr 30454

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 30453	. 2 class Retr
2		vj	. . 3 setvar 𝑗
3		vk	. . 3 setvar 𝑘
4		ctop 20517	. . 3 class Top
5		vr	. . . . . . . . 9 setvar 𝑟
6	5	cv 1474	. . . . . . . 8 class 𝑟
7		vs	. . . . . . . . 9 setvar 𝑠
8	7	cv 1474	. . . . . . . 8 class 𝑠
9	6, 8	ccom 5042	. . . . . . 7 class (𝑟 ∘ 𝑠)
10		cid 4948	. . . . . . . 8 class I
11	2	cv 1474	. . . . . . . . 9 class 𝑗
12	11	cuni 4372	. . . . . . . 8 class ∪ 𝑗
13	10, 12	cres 5040	. . . . . . 7 class ( I ↾ ∪ 𝑗)
14		chtpy 22574	. . . . . . . 8 class Htpy
15	11, 11, 14	co 6549	. . . . . . 7 class (𝑗 Htpy 𝑗)
16	9, 13, 15	co 6549	. . . . . 6 class ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗))
17		c0 3874	. . . . . 6 class ∅
18	16, 17	wne 2780	. . . . 5 wff ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
19	3	cv 1474	. . . . . 6 class 𝑘
20		ccn 20838	. . . . . 6 class Cn
21	19, 11, 20	co 6549	. . . . 5 class (𝑘 Cn 𝑗)
22	18, 7, 21	wrex 2897	. . . 4 wff ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
23	11, 19, 20	co 6549	. . . 4 class (𝑗 Cn 𝑘)
24	22, 5, 23	crab 2900	. . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}
25	2, 3, 4, 4, 24	cmpt2 6551	. 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 1475	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)