Nerve of an open covering
Encyclopedia
In mathematics
, the nerve of an open covering is a construction in topology
, of an abstract simplicial complex
from an open covering of a topological space
X.
The notion of nerve was introduced by Pavel Alexandrov.
Given an index set
I, and open set
s Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:
Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.
In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-sphere with two contractible sets U and V, in such a way that N is a 1-simplex
. However, if we also insist that the open sets corresponding to every intersection
indexed by a set in N is also contractible
, the position changes. This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of N.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the nerve of an open covering is a construction in 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...
, of an abstract simplicial complex
Abstract simplicial complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
from an open covering of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X.
The notion of nerve was introduced by Pavel Alexandrov.
Given an index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
I, and open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:
- the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
belongs to N; - a finite set J ⊆ I belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty. That is, if and only if
Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.
In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-sphere with two contractible sets U and V, in such a way that N is a 1-simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
. However, if we also insist that the open sets corresponding to every intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
indexed by a set in N is also contractible
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point....
, the position changes. This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of N.