In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a development is a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
collection of
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
s of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that satisfies certain
separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
s.
Let
be a topological space. A development for
is a countable collection
of open coverings of
, such that for any closed subset
and any point
in the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of
, there exists a cover
such that no element of
which contains
intersects
. A space with a development is called developable.
A development
such that
for all
is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If
is a
refinement
Refinement may refer to: Mathematics
* Equilibrium refinement, the identification of actualized equilibria in game theory
* Refinement of an equivalence relation, in mathematics
** Refinement (topology), the refinement of an open cover in mathem ...
of
, for all
, then the development is called a refined development.
Vickery's theorem implies that a topological space is a
Moore space if and only if it is
regular and developable.
References
*
*
* {{PlanetMath attribution, id=6495, title=Development
General topology