Definition df-retr 36191

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 36190	. 2 class Retr
2		vj	. . 3 setvar 𝑗
3		vk	. . 3 setvar 𝑘
4		ctop 23266	. . 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 5662	. . . . . . 7 class (𝑟 ∘ 𝑠)
10		cid 5552	. . . . . . . 8 class I
11	2	cv 1569	. . . . . . . . 9 class 𝑗
12	11	cuni 4872	. . . . . . . 8 class ∪ 𝑗
13	10, 12	cres 5660	. . . . . . 7 class ( I ↾ ∪ 𝑗)
14		chtpy 25343	. . . . . . . 8 class Htpy
15	11, 11, 14	co 7451	. . . . . . 7 class (𝑗 Htpy 𝑗)
16	9, 13, 15	co 7451	. . . . . 6 class ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗))
17		c0 4281	. . . . . 6 class ∅
18	16, 17	wne 2959	. . . . 5 wff ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
19	3	cv 1569	. . . . . 6 class 𝑘
20		ccn 23597	. . . . . 6 class Cn
21	19, 11, 20	co 7451	. . . . 5 class (𝑘 Cn 𝑗)
22	18, 7, 21	wrex 3090	. . . 4 wff ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
23	11, 19, 20	co 7451	. . . 4 class (𝑗 Cn 𝑘)
24	22, 5, 23	crab 3416	. . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}
25	2, 3, 4, 4, 24	cmpo 7453	. 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)