Cover
A family gamma of nonempty subsets of X whose union contains the given set X (and which contains no duplicated subsets) is called a cover (or covering) of X. For example, there is only a single cover of {1}, namely {{1}}. However, there are five covers of {1,2}, namely {{1},{2}}, {{1,2}}, {{1},{1,2}}, {{2},{1,2}}, and {{1},{2},{1,2}}.
A minimal cover is a cover for which removal of one member destroys the covering property. For example, of the five covers of {1,2}, only {{1},{2}} and {{1,2}} are minimal covers. There are various other types of specialized covers, including proper covers, antichain covers, k-covers, and k^*-covers (Macula 1994).
In graph theory, a graph cover is a graph that maps locally bijectively onto a base graph. In plainer language, a graph cover is a graph whose edges around each lifted vertex correspond one-to-one with the edges around its image in the base graph.
The number of possible covers for a set of N elements are
the first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ... (OEIS A003465).
See also
Baseball Cover, Covering Map, Edge Cover, Graph Cover, Lebesgue Covering Dimension, Minimal Cover, Open Cover, Proper Cover, Universal Cover, Vertex CoverExplore with Wolfram|Alpha
References
Eppstein, D. "Covering and Packing." https://ics.uci.edu/~eppstein/junkyard/cover.html.Macula, A. J. "Covers of a Finite Set." Math. Mag. 67, 141-144, 1994.Sloane, N. J. A. Sequences A003465/M4024 and A055621 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
CoverCite this as:
Weisstein, Eric W. "Cover." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cover.html