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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
Let
Y ={
Y i }
be a class of subsets of X such that Y
ↆ∪ i Y i for some Y
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
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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