In topology, a retraction, as the name suggests, "retracts" an entire space into a subspace. A deformation retract is a map which captures the idea of continuously shrinking a space into a subspace. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... Screenshot (from SSCX Star Warzone). ... Partial plot of a function f. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Retract

Let X be a topological space and A a subspace of X. Then a continuous map is a retract if the restriction of r to A is the identity map on A; that is, r(a) = a for all a in A. Note that a retraction maps X onto A. In this case, A is called a retract of X, and r is called a retraction. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Partial plot of a function f. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

Neighborhood retract

Let X be a topological space and A a subspace of X. If there exists an open set U such that and A is a retract of U, then A is called a neighborhood retract of X. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Deformation retract

A continuous map is a deformation retract if, for every x in X, a in A, and t in [0, 1],

d(x,0) = x

d(a,1) = a.

A deformation retraction is thus a homotopy between the identity map on X and a retraction of X onto A. A is called a deformation retract of X. The two bold paths shown above are homotopic relative to their endpoints. ...

Note that although homotopy is an equivalence relation between maps, deformation retraction is not an equivalence relation between spaces. Generally one space is a proper subset of the other. However, deformation retraction is related to the notion of homotopy equivalence, in the sense that two spaces are homotopy equivalent if and only if they are both deformation retracts of a single space. This is an equivalence relation. In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

Any topological space which deformation retracts to a point is contractible. Contractibility, however, is a weaker condition, as contractible spaces exist which do not deformation retract to a point. In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ...

References

• This article incorporates material from Neighborhood retract on PlanetMath, which is licensed under the GFDL.