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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 X be a topological space. A development for X is a countable collection F_1, F_2, \ldots of open coverings of X, such that for any closed subset C \subset X and any point p 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 C, there exists a cover F_j such that no element of F_j which contains p intersects C. A space with a development is called developable. A development F_1, F_2,\ldots such that F_\subset F_i for all i is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If F_ is a refinement of F_i, for all i, 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