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) be topological spaces. A function
f
from
X
into
X
is
continuous
iff the inverse image
f
−
1
(
Y
) of every open subset
Y
of
X
is an open subset of
X
, i.e.
) and (
X
,
Definition 2.10
Let (
X
,
T
T
f
−
1
(
Y
)
Y
∈
T
implies
∈
T
.
2
Example 2.8
are con-
tinuous with respect to the usual topologies. For example, consider the projection
ˀ
:
The projection mappings from the plane
R
into the line
R
2
y
. The inverse of any open interval (
a
,
b
)isan
infinite open strip parallel to the
x
-axis.
Continuous functions can also be characterised by their behaviour with respect to
closed sets.
R
ₒ R
defined by
ˀ
(
x
,
y
)
=
Theorem 2.2
Y is continuous iff the inverse image of every
closed subset of Y is a closed subset of X.
A function f
:
X
ₒ
Let
Y
={
Y
i
}
be a class of subsets of
X
such that
Y
ↆ∪
i
Y
i
for some
Y
ↆ
X
.
Then
is called a
cover
of
Y
, and an
open cover
if each
Y
i
is open. Furthermore, if
a finite subclass of
Y
Y
is also a cover of
Y
, i.e. if
∃
Y
i
1
,
...
,
Y
i
m
∈
Y
ↆ
Y
i
1
∪···∪
,
such that
Y
Y
i
m
then
Y
is said to contain a
finite subcover
of
Y
.
Example 2.9
The classical Heine-Borel theorem says that every open cover of a
closed and bounded interval [
a
,
b
] on the real line contains a finite subcover.
Definition 2.11
A subset
Y
of a topological space
X
is
compact
if every open cover
of
Y
contains a finite subcover.
Example 2.10
By the Heine-Borel theorem, every closed and bounded interval
[
a
,
b
] on the real line
R
is compact.
Theorem 2.3
Continuous images of compact sets are compact.
A set
{
X
i
}
of sets is said to have the
finite intersection property
if every finite
subset
{
X
i
1
,
...
,
X
i
m
}
has a nonempty intersection, i.e.
X
i
1
∩···∩
X
i
m
=∅
.
Example 2.11
Consider the following class of open intervals:
(0, 1),
0,
1
2
,
0,
2
2
,
...
1
X
=
Clearly, it has the finite intersection property. Observe however that
X
has an empty
intersection.
Theorem 2.4
of closed subsets
of X that satisfies the finite intersection property has, itself, a nonempty intersection.
A topological space X is compact iff every class
{
X
i
}
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