Graphics Reference
In-Depth Information
3.3
TOPOLOGICAL SPACES
With metric spaces and distances we have the formal definitions needed to measure sizes: topol-
ogy is necessary now to formalize the concept of closeness of points and to study the nature of
spaces in terms of the adjacency between its points. Topological spaces are mathematical struc-
tures that generalize concepts such as closeness, limits, connectedness, or continuity, from the
Euclidean space
R
n
to arbitrary sets of points. is is achieved associating to the space a formal
structure of sub-sets of the space together with their relationships, rather than distances between
points. To deepen the understanding of the concepts presented here, we suggest the following
textbook [
212
].
A
topological space
.X;/
is a set
X
on which a
topology
is defined, that is, a collection of
subsets of
X
, which are called the
open
subsets of
X
and which satisfy the following axioms:
1.
both
X
itself and the empty subset must be among the open sets:
X2
,
;2
;
2.
the intersection of two open sets is open: if
A
,
B2
, then
A\B2
;
3.
all unions of open sets are open: for any collection
A
ii
, if all
A
i
2
, then
[
i
A
i
2
.
e complements in
X
of open sets are called
closed sets
.
ere are two trivial examples of topology we may think of: the coarsest and the finest.
e coarsest topology is the
trivial topology
, which has only two open sets, namely the empty set
and
X
. e finest topology, the
discrete topology
, contains all subsets of
X
as open sets. Another
example, which is intuitive for everyone, is the Euclidean topology defined by intervals. In
R
, an
open interval is a set
.a;b/Dfx2Rja < x < bg
,
a2R[f1g
,
a2R[fC1g
; the topology
defined through the open intervals over
R
is called
Euclidean
topology.
Given a topological space
.X;/
, a
neighborhood
of a point
x2X
is a set containing an
open set which contains
x
. Open sets and distances contribute together to the characterization
of spaces whose points can be separated: a topological space is called a
Hausdorff space
if and only
if every two points
x
,
y
have disjoint neighborhoods.
Metric and topological spaces
e notions of metric and topological spaces are tightly con-
nected. Any space equipped with a metric,
.X;d/
, can be turned into a topological space
.X;
d
/
,
with a quite simple procedure which relies on the metric
d
for building the structure of open sets.
Define the
open ball
of center
x
and radius
r
as the set
B.x;r/Dfy2XWd.x;y/ < rg
,
with
x2X
and
r2R
,
r > 0
. A subset of
X
which is the union of (finitely or infinitely many)
balls is called an
open set
. Equivalently, a subset
U
of
X
is called an
open
set if, given a point
x2U
, there exists a real number
> 0
such that, for any point
y2X
such that
d.x;y/ <
,
y2U
. Now, we can define the topology
d
as the collection of open sets defined as above, then
.X;
d
/
is a topological space with the
metric topology
induced by
d
.
With the previous notation, the open balls of the Euclidean space
R
n
are then defined as
B.x;r/Dfy2R
n
Wjjx;yjj< rg
, with
x2R
n
and
r2R
,
r > 0
.
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