Exterior of A Set
The exterior of a subset S of a topological space X, denoted ext(S) or Ext(S), is the interior int(X \ S) of its relative complement. Alternatively, it can be defined as X \ S—, the complement of the closure of S. Many properties follow in a straightforward way from those of the interior operator, such as the following.
- ext(S) is an open set that is disjoint with S.
- ext(S) is the union of all open sets that are disjoint with S.
- ext(S) is the largest open set that is disjoint with S.
- If S is a subset of T, then ext(S) is a superset of ext(T).
Unlike the interior operator, ext is not idempotent, but the following holds:
- ext(ext(S)) is a superset of int(S).
Read more about this topic: Interior (topology)
Famous quotes containing the words exterior and/or set:
“The competent leader of men cares little for the niceties of other peoples’ characters: he cares much—everything—for the exterior uses to which they may be put.... These are men to be moved. How should he move them? He supplies the power; others simply the materials on which that power operates.”
—Woodrow Wilson (1856–1924)
“When you have come into the land that the LORD your God is giving you, and have taken possession of it and settled in it, and you say, “I will set a king over me, like all the nations that are around me,” you may indeed set over you a king whom the LORD your God will choose. One of your own community you may set as king over you; you are not permitted to put a foreigner over you, who is not of your own community.”
—Bible: Hebrew, Deuteronomy 17:14,15.