# Dictionary Definition

retract

### Verb

1 formally reject or disavow a formerly held
belief, usually under pressure; "He retracted his earlier
statements about his religion"; "She abjured her beliefs" [syn:
abjure, recant, forswear, resile]

2 pull away from a source of disgust or fear
[syn: shrink
back]

3 use a surgical instrument to hold open (the
edges of a wound or an organ) [syn: pull back,
draw
back]

4 pull inward or towards a center; "The pilot
drew in the landing gear"; "The cat retracted his claws" [syn:
draw
in]

# User Contributed Dictionary

## English

### Pronunciation

(US) IPA: /rəˈtrækt/- Rhymes: -ækt

### Verb

# Extensive Definition

In topology, a retraction, as the
name suggests, "retracts" an entire space
into a subspace. A
deformation retraction is a map
which captures the idea of continuously
shrinking a space into a subspace.

## Definitions

### Retract

Let X be a topological
space and A a subspace of X. Then a continuous map

- r:X \to A

is a retraction if the
restriction of r to A is the identity map
on A; that is, r(a) = a for all a in A. Equivalently, denoting
by

- \iota : A \hookrightarrow X

the inclusion, a retraction is a continuous map r
such that

- r \circ \iota = id_A,

that is, the composition of r with the inclusion
is the identity of A. Note that, by definition, a retraction maps X
onto A. A subspace A is
called a retract of X if such a retraction exists. For instance,
any space retracts to a point in the obvious way.

A space X is known as an absolute retract (or AR)
if for every normal space
Y that embeds X as a closed subset, X is a retract of Y.

### Neighborhood retract

If there exists an open set U such
that

- A \subset U \subset X

and A is a retract of U, then A is called a
neighborhood retract of X.

A space X is an absolute neighborhood retract (or
ANR) if for every normal space Y that embeds X as a closed subset,
X is a neighborhood retract of Y.

### Deformation retract and Strong deformation retract

A continuous map

- d:X \times [0, 1] \to X

is a deformation retraction if, for every x in X
and a in A,

- d(x,0) = x, \; d(x,1) \in A ,\quad \mbox \quad d(a,1) = a.

In other words, a deformation retraction is a
homotopy between a retract and the identity map on X. The subspace
A is called a deformation retract of X. A deformation retract is a
special case of homotopy
equivalence.

A retract need not be a deformation retract. For
instance, having a single point as a deformation retract would
imply a space is path connected.

Note: An equivalent definition of deformation
retraction is the following. A continuous map r: X → A is
a deformation retraction if it is a retraction and its composition
with the inclusion is homotopic to the identity map on X. In this
formulation, a deformation retraction carries with it a homotopy between the identity
map on X and itself.

If, in the definition of a deformation
retraction, we add the requirement that

- d(a,t) = a\,

for all t in [0, 1], d is called a strong
deformation retraction. In other words, a strong deformation
retraction leaves points in A fixed throughout the homotopy. (Some
authors take this as the definition of deformation
retraction.)

## Neighborhood deformation retract

A pair (X, A) of spaces in U is an NDR-pair if
there exists a map u:X \rightarrow I such that A = u^ (0) and a
homotopy h:I \times X \rightarrow X such that h(0, x) = x for all x
\in X, h(t, a) = a for all (t, a) \in I \times A, and h(1, x) \in A
for all x \in u^ [ 0 , 1). The pair (h, u) is said to be a
representation of (X, A) as an NDR-pair.

## Properties

Deformation retraction is a particular case of
homotopy equivalence. In fact, two spaces are homotopy
equivalent if and
only if they are both deformation retracts of a single larger
space.

Any topological space which deformation retracts
to a point is contractible
and vice versa. However, there exist contractible spaces which do
not strongly deformation retract to a point.

## References

retract in German: Retraktion

retract in Finnish: Deformaatioretrakti

retract in Italian: Retrazione

retract in Polish: Retrakt deformacyjny

retract in Russian: Деформационный
ретракт

# Synonyms, Antonyms and Related Words

abandon, abjure, abolish, abrogate, annul, assert the contrary,
back, back down, back out,
backwater, belie, cancel, cede, climb down, contest, contradict, contravene, controvert, counter, countermand, counterorder, crawfish out,
cringe, cross, deny, disaffirm, disallow, disannul, disavow, disclaim, disgorge, disown, dispense with, dispose
of, disprove, dispute, do away with, do
without, draw back, draw in, drop, duck, dump, eat crow, eat humble pie,
fall back, flinch,
forgo, forswear, gainsay, get along without, get
rid of, give away, give up, have done with, impugn, invalidate, join issue upon,
kiss good-bye, make a sacrifice, make void, not accept, not admit,
nullify, oppose, override, overrule, palinode, part with, pull back,
pull in, pull out, quitclaim, recall, recant, refuse to admit, refute, relinquish, render up,
renege, renounce, repeal, repudiate, rescind, resign, retreat, retrocede, retrograde, reverse, revoke, sacrifice, set aside, shrink, shy, spare, surrender, suspend, swallow, swear off, take, take back, take issue with,
throw up, unsay, vacate, void, waive, wince, withdraw, write off, yield