Shrinking space

In mathematics, in the field of topology, a topological space 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:
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.