Open Covers of Topological Spaces

An open overs of a topological spaceis a collection of open setssuch that

Example

Letbe the set of real numbers with topology induced by the absolute value metric.

is the set of open ballsThis set of open balls is an open cover because

The setis an open subcover.

An open coveris said to be finer or an improvement on some other open coverif for eachthere is somesuch that

Henceis an improvement of