Graphics Reference
In-Depth Information
Topological properties
of spaces are those properties that are “preserved” by home-
omorphisms. We shall see a number of examples of topological properties in later sec-
tions. A
topologist
is someone who tries to find and analyze topological properties of
spaces. A topologist does not distinguish between homeomorphic spaces. An ellipse
looks the same as a circle. Roughly speaking, two spaces are the same to a topologist
if one can be deformed into the other without any ripping or tearing. The deforma-
tions correspond to a one-parameter family of homeomorphism. In laymen's terms,
topology is “rubber sheet geometry.” At the beginning of the next chapter, Chapter 6,
we shall have a lot more to say about the kinds of questions that topology tries to
answer.
Definition.
Let
X
and
Y
be topological spaces. A map f :
X
Æ
Y
is called an
im-
bedding
if the map f :
X
Æ f(
X
) is a homeomorphism between
X
and the subspace
f(
X
) of
Y
.
Imbeddings are one-to-one maps by definition.
5.3.16. Example.
If
A
is a subspace of
X
, then the inclusion map of
A
into
X
is an
imbedding. If one gives the set of rational numbers the
discrete
topology, then
the inclusion map of this set into the reals
R
with the Euclidean topology is
not
an
imbedding.
Next, we give some limit-related definitions.
Definition.
A
is called an
isolated point
of
A
if it has a neighborhood that contains no other point of
A
except
for
a
. A point
p
Let
A
be a subset of a topological space
X
. A point
a
X
is called a
limit
or
accumulation point
of
A
if every neighbor-
hood of
p
contains a point of
A
different from
p
.
Clearly, every point of a subset
A
in a space
X
is either an isolated or a limit
point. A limit point of
A
that does not belong to
A
is a point in the boundary of
A
.
5.3.17. Theorem.
Let
A
be a subset of a metric space
X
. Every neighborhood of a
limit point of
A
contains infinitely many points of
A
.
Proof.
Easy.
Definition.
A sequence of points
p
n
of a topological space
X
is said to
converge
to
the point
p
in
X
if for each neighborhood
U
of
p
in
X
there is some integer m so that
n ≥ m implies that
p
n
U
.
Many of the properties of convergence of sequences in metric spaces generalize
to the context of convergence in topological spaces, but one typically has to add some
additional hypotheses (such as the space being Hausdorff) on the type of topology one
has. For example, the continuity of a function f defined on a space
X
is not in general
equivalent to f(
p
n
) converging to f(
p
) for all
sequences p
n
that converge to
p
(
sequen-
tial continuity
). One has to allow something more general than sequences. See [Eise74]
and the discussion of
nets
.
