Geography Reference
In-Depth Information
Definition 3.2 (Continuity, homeomorphism).
Let
X
and
Y
be two topological spaces and
f
:
X
→
Y
a mapping.
f
is called
sequence continuous
or
continuous
in
x ∈
X
if
f
to
x
convergent sequences are continuously mapped to
f
(
x
)convergent
series.
f
is called
sequence continuous
or just
continuous
if
f
is continuous in any point
x
of
X
.
f
is called
a topological map
or a
homeomorphism
if
f
is continuous, bijective, and
f
−
1
is continuous
as well.
End of Definition.
1
1
In order to illustrate this in detail, let us here consider the topologies on
S
r
and
φ
(
S
r
)aswellasthe
2
2
topologies on
S
r
and
φ
(
S
r
): compare with Example
3.1
and Fig.
3.1
as well as with Example
3.2
and Fig.
3.2
, respectively.
Fig. 3.1.
Manifold “one-sphere”
S
r
,chart
Φ
(
x
)=
λ
(
x
): the angular parameter is
λ
∈
]0
,
2
π
[
r
, one-dimensional manifold, topology).
Example 3.1 (Circle
S
r
. Second, we present the topology on
φ
(
r
). The “one-sphere”
First, we present the topology on
S
S
r
(circle of radius
r
) is defined as the manifold
S
1
2
x
2
+
y
2
=
r
2
,r
+
,r>
0
1
S
r
:=
{
x
∈
R
|
∈
R
}
,
U
:=
S
r
/
{
x
=+
r
}
.
(3.1)
1
(i) Topology on
S
r
.
r
is defined by the
Euclidean metric
, namely the distance function of the
The topology on
S
ambient space
R
3
, i.e.
d(
x
1
,
x
2
):=
x
1
−
x
2
2
.
(3.2)
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