Shape theory is a branch of the mathematical field of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. Homotopy theory is not appropriate for spaces with bad local properties, hence the need for replacement of homotopy theory by a more sophisticated approach. Thus the overall goal of shape theory is to modify the methods and results from homotopy theory

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 for spaces with good local properties

Local property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.-Properties of a single space:...

, like CW complex

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

es, to more general compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s or compact Hausdorff space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

s with possibly bad local properties.

Shape theory was founded by the Polish mathematician Karol Borsuk

Karol Borsuk

Karol Borsuk was a Polish mathematician.His main interest was topology.Borsuk introduced the theory of absolute retracts and absolute neighborhood retracts , and the cohomotopy groups, later called Borsuk-Spanier cohomotopy groups. He also founded the so called Shape theory...

 in 1968. Borsuk lived and worked in Warsaw

Warsaw

Warsaw is the capital and largest city of Poland. It is located on the Vistula River, roughly from the Baltic Sea and from the Carpathian Mountains. Its population in 2010 was estimated at 1,716,855 residents with a greater metropolitan area of 2,631,902 residents, making Warsaw the 10th most...

, hence the name of one of the fundamental examples of the area, the Warsaw circle. This is a compact subset of the plane produced by "closing up" a topologist's sine curve

Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....

 with an arc. It has homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

s isomorphic

Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...

 to those of a point, but is not homotopy equivalent to it; Whitehead's theorem

Whitehead theorem

In homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....

 does not apply because the Warsaw circle is not a CW complex.

Borsuk's original shape theory has been replaced by a more systematic approach by inverse systems, pioneered by Sibe Mardešić, and independently, by Timothy Porter. In abstract terms, one starts with a dense subcategory of good objects, and approximates general objects by inverse systems of good objects in the best way in the sense of certain universality property. Thus the object is replaced by pro-object in dense category in appropriate way.

For some purposes, like dynamical systems, more sophisticated invariants were developed under the name strong shape. Generalizations to noncommutative geometry

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

, e.g. the shape theory for operator algebra

Operator algebra

In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...

s have been found.