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:
“Professor Eucalyptus said, “The search
For reality is as momentous as
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And the poet’s search for the same exterior made
Interior: breathless things broodingly abreath....”
—Wallace Stevens (1879–1955)
“1st Witch. When shall we three meet again?
In thunder, lightning, or in rain?
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3rd Witch. That will be ere set of sun.
1st Witch. Where the place?
2nd Witch. Upon the heath.
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—William Shakespeare (1564–1616)