A complex K ≤ n ′ which can be obtained from a complex K ≤ n by a finite sequence of retractions is called a retract of the complex K ≤ n.
The set is a retract of in if there exists a retraction from to in.
[ 11 ] A homomorphism r of a lattice L onto a lattice M is said to be a retraction if there exists a homomorphism s of M into L which satisfies r∘s=i d M. A lattice M is called a retract of a lattice L if there is a retraction r, of L onto M, and s is then called a pseudo-inverse of r.
Note that AR ⊂ ANR ⊂ AANR.12 Observe also that if A is a retract of a topological space X with retraction (r:Xrightarrow A), then any continuous mapping (f_{0} Arightarrow Y) into any topological space Y extends to the continuous mapping (f=rcirc f_{0}:Xrightarrow Arightarrow Y).
If A ⊂ B are subsets of a topological space and π : B → A is a continuous mapping from ℬ onto A such that π ( p ) = p for every p ∈ A, then π is said to be a retraction of ℬ onto A. When a retraction of ℬ onto A exists, A is called a retract of ℬ.
There are two vertices u, v and a retraction taking u to v and a subcomplex K ≤ n ′ being a retract of K ≤ n.
Moreover, any amenable discrete group is a retract of a discrete C∗-unique group.
Let Z ⊂ W ∪ ω ∗ be a given set such that Z ∩ W is a retract of W but not a retract of Z.
If (V=X), A is simply said to be a retract of X.11.
Then has a fixed point in and is a retract of.
