In
mathematics, in the 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 ho ...
, 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 ...
is said to be a shrinking space if every
open cover admits a shrinking. A ''shrinking'' of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding original open set.
[.]
Properties
The following facts are known about shrinking spaces:
* Every shrinking space is
normal.
* Every shrinking space is
countably paracompact.
* In a
normal space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space ...
, every locally finite, and in fact, every point-finite open cover admits a shrinking.
* Thus, every normal
metacompact space is a shrinking space. In particular, every
paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...
is a shrinking space.
These facts are particularly important because shrinking of open covers is a common technique in the theory of
differential manifolds and while constructing functions using a
partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are ...
.
See also
*
References
* General topology, Stephen Willard, definition 15.9 p. 104
Topology
Properties of topological spaces
Topological spaces
{{topology-stub