- The intersection of a finite number of open subsets is open.
- The union of a finite number of closed sets of a metric space is closed.
- The intersection of any collection of closed sets is closed.
The proofs of (1) and (2) are easily made by applying Definition 2.12.
Properties 3 and 4 follow from (2) and(l), respectively, by applying De
Morgan's laws (Theorem 2.1-11).
Theorem 2.3 The following properties hold:- Let (S,d) be a metric space and let (M,dM) be a nonempty subspace
with the relative metric. A subset E of M is open in (M, dM) iff it is
of the form E = Mn A, where A is open in (S, d).
- A subset F of Mis closed in (M, dM) iff it is of the form F =Mn B,
where B is closed in (S, d). - 7 SJETS ASSOCIATED WITH A GIVEN SET
Although not every subset of a metric space ( S, d) is either open or closed,
it is possible to associate with each subset A C S a corresponding open
set and a corresponding closed set. There are also some other sets that
are related to a given set A.Definition 2.14 The interior of a set A C S is the largest open set
contained in A. The interior of A is denoted by Int A, and also by A^0 •
Alternatively, the interior of A is the union of all open sets contained in
A, or the set of all points of A that have s.ome 6-neighborhood ~ontained
in A. Also, it may be characterized simply by saying that it is the set of
all points of which A is a neighborhood. It may happen that Int A = 0
when the conditions above are vacuously satisfied.Definiti.on 2.15 The closure of a set A C Sis the smallest closed set con-
taining A. The closure of A is denoted by A, A - , or by Cl A.· Alternatively,
the closure of A is the intersection of all supersets of A, or else
A={x: No(x)nAf0 for every o > O}
Definiti.on 2.16 The exterior of a set A C S is the interior of the
complement of A. The exterior of A is denoted Ext A. Hence we have
Ext A = Int A'. Alternatively, Ext A = (A)'.
Definition 2.17 The boundary of a set A C S, denoted 8A or Bd A, is
defined to be its closure minus the interior of A, i.e., 8A = A - Int A.
Equivalently,
8A={x: N5(x)nAf0,N5(x)nA'f0 for every o > O}