In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a development is a
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
collection of
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family 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_\alpha\su ...
s of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ...
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 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