adj. 1a. (of a topological space) having the property that every collection of open sets the union of which is the whole space has a finite subcollection with the same property. This is equivalent to the finite intersection property. In particular, in n-dimensional Euclidean space a set is compact if and only if it is closed and bounded. For example, the closed interval [0, 1] is compact, but the open interval (0, 1) is not, as
{(1/n, 1)}_n∈N
is a cover of (0, 1) that has no finite subcover. See also Lindelöf space. Compare sequentially compact.
b. (of a subspace) such that every cover in the induced topology has a finite subcover. 2. (of a relation) having the property that for any pair of elements such that a is related to b, there is some element c such that a is related to c and c is related to b. For example, less than is compact on the rational numbers, since for any pair of rationals, a and b, 1/2(a + b) is a rational between a and b. 3. (of a mapping between topological vector spaces, especially Banach spaces) having the property that the image of any bounded set has a compact closure. See also completely continuous.