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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-retr | Structured version Visualization version Unicode version |
Description: Define the set of retractions on two topological spaces. We say that is a retraction from to . or Retr iff there is an such that are continuous functions called the retraction and section respectively, and their composite is homotopic to the identity map. If a retraction exists, we say is a retract of . (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.) |
Ref | Expression |
---|---|
df-retr | Retr Htpy |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cretr 31199 | . 2 Retr | |
2 | vj | . . 3 | |
3 | vk | . . 3 | |
4 | ctop 20698 | . . 3 | |
5 | vr | . . . . . . . . 9 | |
6 | 5 | cv 1482 | . . . . . . . 8 |
7 | vs | . . . . . . . . 9 | |
8 | 7 | cv 1482 | . . . . . . . 8 |
9 | 6, 8 | ccom 5118 | . . . . . . 7 |
10 | cid 5023 | . . . . . . . 8 | |
11 | 2 | cv 1482 | . . . . . . . . 9 |
12 | 11 | cuni 4436 | . . . . . . . 8 |
13 | 10, 12 | cres 5116 | . . . . . . 7 |
14 | chtpy 22766 | . . . . . . . 8 Htpy | |
15 | 11, 11, 14 | co 6650 | . . . . . . 7 Htpy |
16 | 9, 13, 15 | co 6650 | . . . . . 6 Htpy |
17 | c0 3915 | . . . . . 6 | |
18 | 16, 17 | wne 2794 | . . . . 5 Htpy |
19 | 3 | cv 1482 | . . . . . 6 |
20 | ccn 21028 | . . . . . 6 | |
21 | 19, 11, 20 | co 6650 | . . . . 5 |
22 | 18, 7, 21 | wrex 2913 | . . . 4 Htpy |
23 | 11, 19, 20 | co 6650 | . . . 4 |
24 | 22, 5, 23 | crab 2916 | . . 3 Htpy |
25 | 2, 3, 4, 4, 24 | cmpt2 6652 | . 2 Htpy |
26 | 1, 25 | wceq 1483 | 1 Retr Htpy |
Colors of variables: wff setvar class |
This definition is referenced by: (None) |
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