In a paracompact Hausdorff space X \,, if \ mathcal { O } \, is an open cover, then there exists a partition of unity subordinate to it.
32.
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.
33.
For a cohomology class represented by closed form in, express as a difference of forms via a partition of unity subordinate to the open cover } }, for example.
34.
I know that a space X is Lindel�f if every open cover has a countable subcover and that X is a second countable space of its topology has a countable basis.
35.
Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.
36.
Commonly seen in industrial applications such as the pictured pneumatic cylinder, they are also used in consumer equipment; for example some computer printers use them to detect missing paper and open covers.
37.
In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of " A " for which the limit of the lengths of the covers is zero.
38.
Therefore, there is a countable collections of open intervals in which is an open cover of " X " " ? ", such that the sum over all their lengths is arbitrarily small.
39.
These interiors consist of a finite open cover of " X " 1 / " n ", possibly up to a finite number of points ( which may fall on interval edges ).
40.
Where \ check H ^ q ( \ mathcal U, \ mathcal F ) is the q-th ech cohomology group of \ mathcal F with respect to the open cover \ mathcal U.
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