Definition df-retr 35749

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 35748	. 2 class Retr
2		vj	. . 3 setvar 𝑗
3		vk	. . 3 setvar 𝑘
4		ctop 23102	. . 3 class Top
5		vr	. . . . . . . . 9 setvar 𝑟
6	5	cv 1569	. . . . . . . 8 class 𝑟
7		vs	. . . . . . . . 9 setvar 𝑠
8	7	cv 1569	. . . . . . . 8 class 𝑠
9	6, 8	ccom 5667	. . . . . . 7 class (𝑟 ∘ 𝑠)
10		cid 5557	. . . . . . . 8 class I
11	2	cv 1569	. . . . . . . . 9 class 𝑗
12	11	cuni 4874	. . . . . . . 8 class ∪ 𝑗
13	10, 12	cres 5665	. . . . . . 7 class ( I ↾ ∪ 𝑗)
14		chtpy 25179	. . . . . . . 8 class Htpy
15	11, 11, 14	co 7419	. . . . . . 7 class (𝑗 Htpy 𝑗)
16	9, 13, 15	co 7419	. . . . . 6 class ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗))
17		c0 4286	. . . . . 6 class ∅
18	16, 17	wne 2960	. . . . 5 wff ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
19	3	cv 1569	. . . . . 6 class 𝑘
20		ccn 23433	. . . . . 6 class Cn
21	19, 11, 20	co 7419	. . . . 5 class (𝑘 Cn 𝑗)
22	18, 7, 21	wrex 3091	. . . 4 wff ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
23	11, 19, 20	co 7419	. . . 4 class (𝑗 Cn 𝑘)
24	22, 5, 23	crab 3418	. . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}
25	2, 3, 4, 4, 24	cmpo 7421	. 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 1570	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 used by: (None)