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:
“It’s not a pretty face, I grant you. But underneath its flabby exterior is an enormous lack of character.”
—Alan Jay Lerner (1918–1986)
“Who set this ancient quarrel new abroach?”
—William Shakespeare (1564–1616)