A cover (or covering) of a set 
is a family 
of distinct nonempty subsets of 
whose union is 
. For example, there is only a single cover of 
, namely 
.
However, there are five covers of 
, namely 
, 
,
, 
, and 
.
A minimal cover is a cover for which removal of any one member destroys the covering property. For example, of the five covers of
, only 
and 
are minimal covers.
There are various other types of specialized covers, including proper
covers, antichain covers, 
-covers, and 
-covers (Macula 1994).
In graph theory, a graph cover is a graph that maps locally bijectively onto a base graph. For simple graphs, the edges incident to each graph vertex of the covering graph correspond one-to-one with the edges incident to its image in the base graph.
The number 
of covers of a labeled set of 
elements, for an integer 
, is
 |
giving 1, 5, 109, 32297, 2147321017, 9223372023970362989, ... for 
, 2, ... (OEIS A003465).
The empty family is the unique cover of the empty set,
so 
.
If covers that differ only by relabeling the underlying set are identified, the counts for 
,
2, ... are instead 1, 4, 34, 1952, ... (OEIS A055621).
