In the mathematical field of topology, a development is a countable collection of open covers of a topological space that satisfies certain separation axioms.
Let
X
be a topological space. A development for
X
is a countable collection
F
1
,
F
2
,
…
of open coverings of
X
, such that for any closed subset
C
⊂
X
and any point
p
in the complement 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
,
…
such that
F
i
+
1
⊂
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
i
+
1
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.
