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 exterior must be joined to the interior to obtain anything from God, that is to say, we must kneel, pray with the lips, and so on, in order that proud man, who would not submit himself to God, may be now subject to the creature.”
—Blaise Pascal (1623–1662)
“Speak of me as I am. Nothing extenuate,
Nor set down aught in malice.”
—William Shakespeare (1564–1616)