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
{\displaystyle X}
be a topological space. A development for
X
{\displaystyle X}
is a countable collection
F
1
,
F
2
,
…
{\displaystyle F_{1},F_{2},\ldots }
of open coverings of
X
{\displaystyle X}
, such that for any closed subset
C
⊂
X
{\displaystyle C\subset X}
and any point
p
{\displaystyle p}
in the complement of
C
{\displaystyle C}
, there exists a cover
F
j
{\displaystyle F_{j}}
such that no element of
F
j
{\displaystyle F_{j}}
which contains
p
{\displaystyle p}
intersects
C
{\displaystyle C}
. A space with a development is called developable.
A development
F
1
,
F
2
,
…
{\displaystyle F_{1},F_{2},\ldots }
such that
F
i
+
1
⊂
F
i
{\displaystyle F_{i+1}\subset F_{i}}
for all
i
{\displaystyle 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
{\displaystyle F_{i+1}}
is a refinement of
F
i
{\displaystyle F_{i}}
, for all
i
{\displaystyle 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.