open cover
An open cover is a way of blanketing a space with open patches so that no point is left exposed. Picture spreading overlapping open umbrellas over a region until every spot is under at least one umbrella — the collection of umbrellas is an open cover. The patches may overlap freely and there may be infinitely many of them.
Formally, an open cover of a topological space X (or of a subset A) is a family {U_i} of open sets whose union contains X (respectively contains A). Each U_i is one open set of the cover; the only requirement is that the union of the whole family leaves nothing out.
Open covers are the raw material of compactness: the question “does every open cover have a finite subcover?” is exactly what compactness answers. They also appear in definitions of paracompactness, in partitions of unity, and throughout analysis whenever a local property (true near each point) must be promoted to a global one (true on all of X).
The intervals (n - 1, n + 1) for every integer n form an open cover of the whole real line: each point lies in at least one (indeed two) of them, and together they cover everything. No finite subfamily covers the line, which reflects that the line is not compact.
Unit intervals centered at the integers cover the line; no finite selection can, since the line is unbounded.