Graphics Reference
In-Depth Information
Definition. Let A be a subset of a topological space X . A boundary point of A is a
point p in X such that every neighborhood of p meets both A and the complement of
A , X - A . The boundary of A (in X ), denoted by bdry( A ), is defined to be the set of
boundary points of A . A point a in A is called an interior point of A (in X ) if a has a
neighborhood U in X that is contained in A . The set of interior points of A is called
the interior of A and is denoted by int( A ). The closure of A (in X ), denoted by cls( A ),
is defined to be the set of all points p in X with the property that every neighborhood
of p meets A .
These definitions are analogous to those in Section 4.2. It is important to note
that the above definitions of boundary, interior, and closure are all relative to the con-
taining space X . One can show (Exercise 5.3.2) that
() =-
()
int AA
bdry
A
,
(5.6)
() =
() »
()
cls
A
int
A
bdry
A
.
(5.7)
Definition. A subset A of a topological space X is said to be dense in X if it inter-
sects every nonempty open set of X . It is said to be nowhere dense in X if its closure
contains no nonempty open subset of X .
For example, the rational numbers are dense in R , as are the irrational numbers.
Any finite subset of R n is nowhere dense in R n .
5.3.7. Lemma. Let ( X ,T) be a topological space and let Y be a subset of X . Define
a set S of subsets of Y by
{
}
S =«
AYAT .
Œ
(5.8)
Then S is a topology on Y .
Proof.
The proof is straightforward.
Definition. Let ( X ,T) be a topological space and let Y be a subset of X . The topol-
ogy S on Y defined by equation (5.8) is called the relative topology on Y induced by T .
The topological space ( Y ,S) is called a subspace of ( X ,T) . If T is clear from the context,
one simply talks about the subspace Y of X .
“Induced metrics” and “relative topologies” may sound abstract, but these
notions tend to get used automatically in the context of subspaces of R n , although
without explicitly using those terms. For example, one probably had no hesitation to
talk about the distance between points of a circle or open arcs in a circle .
5.3.8. Lemma. Let X be a topological space and let Y be a subspace of X . The closed
sets of Y are just the sets A « Y , where A is closed in X .
Proof.
This is an easy consequence of the definitions.
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