In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rn (or a manifold with countable base) is an increasing sequence of compact sets, where by increasing we mean is a subset of, with the limit (union) of the sequence being E.
Sometimes one requires the sequence of compact sets to satisfy one more property— that is contained in the interior of for each . This, however, is dispensed in Rn or a manifold with countable base.
For example, consider a unit open disk and the concentric closed disk of each radius inside. That is let and . Then taking the limit (union) of the sequence gives E. The example can be easily generalized in other dimensions.
Famous quotes containing the words exhaustion, compact and/or sets:
“The becoming of man is the history of the exhaustion of his possibilities.”
—Susan Sontag (b. 1933)
“The Puritans, to keep the remembrance of their unity one with another, and of their peaceful compact with the Indians, named their forest settlement CONCORD.”
—Ralph Waldo Emerson (1803–1882)
“The believing mind reaches its perihelion in the so-called Liberals. They believe in each and every quack who sets up his booth in the fairgrounds, including the Communists. The Communists have some talents too, but they always fall short of believing in the Liberals.”
—H.L. (Henry Lewis)