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4 | Basis, Subbasis,
4 | Subspace
Basis, Subbasis,
Subspace
4 | Basis, Subbasis,
Our main goal inSubspace
this chapter is to develop some tools that make it easier to construct examples of
topological spaces. By Definition 3.12 in order to define a topology on a set X we need to specify
which subsets of X are open sets. It can difficult to describe all open sets explicitly, so topological
Our
main
this chapter
is toeither
develop
someortools
that make
easier to Interesting
construct examples
of
spaces
aregoal
oftenindefined
by giving
a basis
a subbasis
of aittopology.
topological
topological
spaces.
By
Definition
3.12
in
order
to
define
a
topology
on
a
set
X
we
need
to
specify
spaces can be also obtained by considering subspaces of topological spaces. We explain these notions
which
below. subsets of X are open sets. It can difficult to describe all open sets explicitly, so topological
spaces are often defined by giving either a basis or a subbasis of a topology. Interesting topological
Our Definition.
main
in
chapter
to let
develop
tools
make itof
easier
tocollection
construct
examples
of
spaces
cangoal
be also
obtained
by is
considering
of that
topological
spaces.
We
explain these
4.1
Letthis
X be
a set
and
B besubspaces
asome
collection
of subsets
X . The
B
is anotions
basis
topological
spaces.the
Byfollowing
Definition
3.12 in order to define a topology on a set X we need to specify
below.
on
X if it satisfies
conditions:
� of X are open sets. It can difficult to describe all open sets explicitly, so topological
which subsets
1) X = V ∈B V ;
spaces
are oftenLet
defined
giving
a basis
or a subbasis
of a of
topology.
InterestingBtopological
4.1
Definition.
X be by
a set
and either
let B be
a collection
of subsets
X . The collection
is a basis
�
V
∈
B
and
�
∈
V
∩
V
there
exists
W
∈
B
such
that
�
∈
W
and W these
⊆ V1 ∩
V2 .
2)
for
any
V
1
2
1
2
spaces
be also the
obtained
by considering
notions
on
X if can
it satisfies
following
conditions: subspaces of topological spaces. We explain
�
below.
1) X = V ∈B V ;
2) for any V � V ∈ B and � ∈ V1 ∩X V2 there exists W ∈ B such that � ∈ W and W ⊆ V1 ∩ V2 .
4.1 Definition. 1Let2X be a set and let
B be a collection of subsets of X . The collection B is a basis
V1
V2
W
on X if it satisfies the following conditions:
�
�
X
1) X = V ∈B V ;
exists
2) for any V1 � V2 ∈ B and � ∈ V1 ∩ V2 Vthere
W WV2∈ B such that � ∈ W and W ⊆ V1 ∩ V2 .
1
X
�
4.2 Example. If (X � �) is a metric space then the set B = {B(�� �) | � ∈ X � � > 0} consisting of all
open balls in X is a basis on X (exercise).V1
V2
W
4.2 Proposition.
Example. If (X
� �)X isbea ametric
space
then
4.3
Let
set, and
let B
be
open
balls
X isbea obtained
basis on X
U ⊆X
thatincan
as(exercise).
the union of
Then T is a topology on X .
4.3 Proposition. Let X be a set, and let B be
4.2
If (X
is a metric
space
then
U ⊆Example.
X that can
be� �)
obtained
as the
union
of
open balls
X is a basis
Then
T is aintopology
on X on
. X (exercise).
�
the
set on
B=
| � ∈ Xthe
� � > 0} consisting of all
a basis
X .{B(��
Let T�)denote
� collection of all subsets
some elements of B, U = V ∈B1 V for some B1 ⊆ B.
a basis on X . Let T denote �
the collection of all subsets
the
setelements
B = {B(��
�) |U� =
∈ X V� ∈B
�>
some
of B,
V0}forconsisting
some B1 of
⊆ all
B.
1
26
4.3 Proposition. Let X be a set, and let B be a basis on X . Let T denote �
the collection of all subsets
U ⊆ X that can be obtained as the union of some
26elements of B, U = V ∈B1 V for some B1 ⊆ B.
Then T is a topology on X .
26
29
4.2 Example. If (X � �) is a metric space then the set B = {B(�� �) | � ∈ X � � > 0} consisting of all
open balls in X is a basis on X (exercise).
4.3 Proposition. Let X be a set, and let B be a basis on X . Let T denote �
the collection of all subsets
U ⊆ X that can be obtained as the union of some elements of B, U = V ∈B1 V for some B1 ⊆ B.
4. Basis, Subbasis, Subspace
27
Then T is a topology on X .
Proof. Exercise.
26
4.4 Definition. Let B be a basis on a set X and let T be the topology defined as in Proposition 4.3.
In such case we will say that B is a basis of the topology T and that T is the topology defined by27
the
basis B.
4. Basis, Subbasis, Subspace
Proof. Exercise.
4.5 Example. Consider R� with the Euclidean metric �. Let B be the collection of all open balls
B(�� �) ⊆ R� such that � ∈ Q and � = (� � � � � � � � � ) where � � � � � � �� ∈ Q. Then B is a basis of the
4.4 Definition. Let B be �a basis on a set1 X2 and let� T be the 1topology
defined as in Proposition 4.3.
Euclidean topology on R (exercise).
In such case we will say that B is a basis of the topology T and that T is the topology defined by the
basis B.
4.6 Basis,
Example.Subbasis,
The set B =Subspace
{[�� �) | �� � ∈ R} is a basis of a certain topology on R. We will call
it
4.
27
the arrow topology.
4.5 Example. Consider R� with the Euclidean metric �. Let B be the collection of all open balls
B(�� �) ⊆ R� such that � ∈ Q and � = (�1 � �2 � � � � � �� ) where �1 � � � � � �� ∈ Q. Then B is a basis of the
Proof.
Exercise.
4.
Basis,
Subbasis,
27
R
Euclidean
topology
on R� Subspace
(exercise).
�
�
4.4 Definition. Let B be a basis on a set X and let T be the topology defined as in Proposition 4.3.
4.6 Example. The set B = {[�� �) | �� � ∈ R} is a basis of a certain topology on R. We will call it
In such
case Subbasis,
we will say that
B is a basis of the topology T and that T is the topology defined by27
the
Proof.
Exercise.
4.
Subspace
the Basis,
arrow topology.
4.7
Example.
Let B = {[�� �] | �� � ∈ R}. The set B is a basis of the discrete topology on R (exercise).
basis
B.
4.4 Definition. Let B be a basis on a set X and let T be the topology defined as in Proposition 4.3.
���with
4.8
Example.
X say
= {��
��
let ofB the
= {{��
�� �}�
��R
The
set topology
B is not
a
basis
ofballs
any
4.5
Consider
R
Euclidean
metric
�. T{��
Let
B�}}.
be T
the
of all
openby
Proof.
Exercise.
In such
case weLet
will
that
B �}
isthe
aand
basis
topology
and
that
is collection
the
defined
the
�
topology
on
X
since
�
∈
{��
��
�}
∩
{��
��
�},
and
B
does
not
contain
any
subset
W
such
that
�
∈
W
B(��
�)
⊆
R
such
that
�
∈
Q
and
�
=
(�
�
�
�
�
�
�
�
�
)
where
�
�
�
�
�
�
�
∈
Q.
Then
B
is
a
basis
of
the
� �
�
1� 2
1
basis B.
�
and
W
⊆
{��
��
�}
∩
{��
��
�}.
Euclidean
topology
onbe
R a (exercise).
4.4
Definition.
Let B
basis on a set X and let T be the topology defined as in Proposition 4.3.
�
In such
case we
will
say{[��
that
B
a∈basis
thesettopology
TLet
andofBthat
T
the topology
topology
defined
the
4.5
Example.
Consider
R
the
Euclidean
metric
be discrete
theis collection
of on
all
�byballs
4.7
Example.
Let
B=
�]with
| ��is�
R}. of
The
B is a�.basis
the
Ropen
(exercise).
4.9
set
S� �be
any
collection
of
subsets
of
X
such
that
X
=
Vit.
4.6
Example.
TheLet
setXB
=aQ
{[��
�)and
��let
� 1∈
R}
is
a
basis
of
a
certain
topology
on
R.
We
will
call
V
∈S
basis
B(��Proposition.
�)B.
⊆ R� such
that
�be
∈
and
�| =
(�
�
�
�
�
�
�
)
where
�
�
�
�
�
�
�
∈
Q.
Then
B
is
a
basis
of
the
�
�
2
1
� of all subsets of X that can be obtained using two operations:
Let
T
denote
the collection
the
arrow
topology.
Euclidean
topology
4.8
Example.
Let X on
=R
{�� (exercise).
�� �� �} and let B = {{�� �� �}� {�� �� �}}. The set B is not a basis of any
� with the Euclidean metric �. Let B be the collection of all open balls
4.5
Example.
Consider
R
1) taking
of ∩sets
S; and B does not contain any subset W such that � ∈ W
topology
on Xfinite
sinceintersections
� ∈ {�� �� �}
{�� in
�� �},
� such that � ∈ Q and � = (� � � � � � � � � ) where � � � � � � � ∈ Q. Then B is a basis of the
B(��
�)taking
⊆
4.6
The
set{��
B��=�}.
{[��
| ��
� 1∈ R}
is 1).
a�basis of a1 certain
� topology on R. We will call it
2 in
and2)Example.
W
⊆R
{��arbitrary
��
�} ∩
unions
of �)
sets
obtained
R
Euclidean
topology on R� (exercise).
the arrow topology.
�
�
�
Then T is a topology on X .
4.9 Proposition. Let X be a set and let S be any collection of subsets of X such that X = V ∈S V .
4.6
Example.
Thecollection
set B = of
{[��
| �� � ∈ofR}
is a basis
a certainusing
topology
on R. We will call it
Let T
denote the
all�)subsets
X that
can beofobtained
two operations:
R the discrete
4.7
Example.
Let B = {[�� �] | �� � ∈ R}. The set B is a basis of
topology on R (exercise).
Proof.
Exercise.
the
arrow
topology.
�
1) taking finite intersections of sets in �S;
� any
4.8
LetLet
X=
{��
��
and
�� �}� {��of�� subsets
�}}. Theofset
B is not
2)Example.
taking arbitrary
of �}
sets
obtained
in{{��
1).collection
4.10
Definition.
Xunions
be ��
a set
and
letletS B
be=any
X such
thataXbasis
= of
V ∈S V .
R the discrete
topology
on XTLet
since
�∈
{��
��| ��
�}�∩∈
{��
�� �},
and
Bthe
does
not contain
any subset
W
such
that
� ∈ WS
4.7
Example.
B
=
{[��
�]
R}.
The
set
is
a
basis
of
topology
on
R
(exercise).
The
topology
defined
by
Proposition
4.9
is
called
topology
generated
by
S,
and
the
collection
Then T is a topology on X .
�
�
and
W ⊆a{��
�� �} ∩ of
{��T.�� �}.
is
called
subbasis
4.8 Example. Let X = {�� �� �� �} and let B = {{�� �� �}� {�� �� �}}. The set B is not a basis
�of any
Proof.
Exercise.
4.9
Proposition.
Let
X
be
a
set
and
let
S
be
any
collection
of
subsets
of
X
such
that
X
=
Vis.
4.7
Example.
Let
B
=
{[��
�]
|
��
�
∈
R}.
The
set
is
a
basis
of
the
discrete
topology
on
R
(exercise).
topology
on
X
since
�
∈
{��
��
�}
∩
{��
��
�},
and
B
does
not
contain
any
subset
W
such
that
�V ∈
∈S
4.11 Example. If X = {�� ��
and S = {{�� �� �}� {�� �� �}} then the topology generated by
SW
Let
T
denote
the
collection
of
all
subsets
of
X
that
can
be
obtained
using
two
operations:
and
W
⊆
{��
��
�}
∩
{��
��
�}.
�
T = {{�� �� �}� {�� �� �}� {���}� {�� �� �� �}� ?}.
4.10
Definition.
Let
X {��
be ��
a set
and
letletS B
be=any
collection
of�� subsets
ofset
X such
thataXbasis
= of
V.
V ∈S
4.81)Example.
Let
X
=
��
�}
and
{{��
��
�}�
{��
�}}.
The
B
is
not
any
taking finite
intersections
of sets 4.9
in S;
�
The
topology
T
defined
by
Proposition
is
called
the
topology
generated
by
S,
and
the
collection
S
topology
on Xarbitrary
since
�Xunions
∈
{��
�}
��S�},
Bcollection
does
not of
contain
Wthat
such
that
4.9
Let
and
be and
any
subsets
ofsubset
Xa basis
such
= �V ∈
Vof.
4.12
Proposition.
Let
(Xbe
� TaX��
),set
�∩TY{��
)let
be
topological
spaces,
and
let any
B be
(or
aXsubbasis)
∈S W
2)Proposition.
taking
of(Y
sets
obtained
in 1).
is
called
a
subbasis
of
T.
and
Wdenote
⊆ {�� the
��� �}
{��Y��is�}.
Let
of
all subsets
be every
obtained
T
A
function
: collection
X∩→
continuous
iff of
� −1X(Vthat
) ∈ can
TX for
V ∈using
B. two operations:
Y. T
Then T is a topology on X .
1) Example.
taking finite
of and
sets Sin=S;{{�� �� �}� {�� �� �}} then the topology generated�by S is
4.11
If Let
Xintersections
=X{��
�}
4.9 Proposition.
be��
a ��
set
and let
S be any collection of subsets of X such that X = V ∈S V .
2)
taking
arbitrary
unions
of
sets
in 1).
T
�� �}�
{��collection
�� �}� {���}�
{��subsets
��obtained
�� �}�of?}.
Let=T{{��
denote
the
of all
X that can be obtained using two operations:
Proof.
Exercise.
Then T is a topology on X .
30
1) Proposition.
taking finite intersections
sets
in topological
S;
�
4.12
Let (X � T ), (Yof� T
) be
spaces, and let B be a basis (or a subbasis)
of
4.10 Definition. Let X be aX set andY let S be
any
collection
of
subsets
of
X
such
that
X
=
V
V ∈S .
arbitrary
of sets obtained
in)1).
TY 2)
. A taking
function
� : X →unions
Y is continuous
iff � −1 (V
∈ TX for every V ∈ B.
The
topology
T defined by Proposition 4.9 is called the topology generated by S, and the collection S
Proof.
Exercise.
Then
T
is
a
topology
is called a subbasis ofonT.X .
R
�
B(��Basis,
�) ⊆ R Subbasis,
such
�∈
Q
andon
�=
(�1 �X�2 �and
� � � �let
�� )Twhere
� � � � � �� ∈defined
Q. Then
a basis of 27
the
4.
4.4
Definition.
Letthat
B be
aSubspace
basis
a set
be the�1topology
as B
in isProposition
4.3.
�
�
� (exercise).
Euclidean
topology
In such case
we will on
sayRthat
B is a basis of the topology T and that T is the topology defined by the
basis B.
Proof.
Exercise.
4.7
Example.
Let
=B
{[��
| ���)� |∈��R}.
is a basis
of the discrete
topology
(exercise).
4.6
TheBset
=�]{[��
� ∈The
R} set
is aBbasis
of a certain
topology
on R. on
WeRwill
call it
�
the
arrow
topology.
4.5 Example. Consider R with the Euclidean metric �. Let B be the collection of all open balls
4.4
LetX
B=be
a ��
basis
a set
XB and
let ��
T �}�
be the
topology
defined
4.3.
4.8
Example.
Let
��and
�}on
let
= {{��
{�� ��
�}}. The
set B as
is in
notProposition
a basis of any
B(��Definition.
�) ⊆ R� such
that
�{��
∈
Q
�and
=
(�
1 � �2 � � � � � �� ) where �1 � � � � � �� ∈ Q. Then B is a basis of the
In
such
case
wesince
will on
say
B�}
is ∩a {��
basis
of the
that Tany
is the
topology
defined
the
�
topology
ontopology
XSubbasis,
�
∈Rthat
{��
��
�� �},
andtopology
B does T
notand
contain
subset
W such
that �by∈27
W
4.
Basis,
Subspace
Euclidean
(exercise).
R
basis
and WB.⊆ {�� �� �} ∩ {�� �� �}.
�
�
4.6 Example. The set B =� {[�� �) | �� � ∈ R} is a basis of a certain topology on R. We will
� call it
4.5
Example.
Consider
R awith
the Euclidean
metric
�. LetofBsubsets
be theofcollection
of all
openV balls
4.9
Proposition.
Let X be
set and
let S be any
collection
X such that
X =
Proof.
Exercise.
∈S V .
the arrow
topology.
4.7
Let collection
B
= {[��
�� �subsets
The
abe
basis
discrete
topology
on
R
(exercise).
B(��
⊆ R� such
that
� ∈ �]
Q
�∈=R}.
(�1 �of
�2X
� � set
�that
� � �B�can
)iswhere
�1 �of� �the
� � �using
∈
Q.
Then
B
is
a
basis
of
the
Let Example.
T�)denote
the
of|and
all
obtained
two
operations:
�
�
Euclidean
topology
onbe
R a (exercise).
4.4
Letintersections
B
basisofonsets
a set
X and let T be the topology defined as in Proposition 4.3.
1)Definition.
taking finite
in S;
4.8
Example.
Let
X
=
{��
��
��
�}
and
let
= {{��
�� �}�T{��
��R�}}.
set topology
B is not a
basis by
of any
In such case we will say that B is a basis ofB the
topology
and
that TThe
is the
defined
the
2)
taking
arbitrary
unions
of
sets
obtained
in
1).
�
�
topology
on
X
since
�
∈
{��
��
�}
∩
{��
��
�},
and
B
does
not
contain
any
subset
W
such
that
�
∈ Wit
4.6
The set B = {[�� �) | �� � ∈ R} is a basis of a certain topology on R. We will call
basisExample.
B.
and arrow
WT ⊆
�� �} ∩ {��
Then
is {��
a topology
on ��
X .�}.
the
topology.
4.7 Example. Let
B = {[��
| �� �
∈ Euclidean
R}. The setmetric
B is a�.basis
topology
(exercise).
4.5
Consider
R��]with
the
Let ofBthe
be discrete
the collection
of on
allRopen
� balls
�
4.9
Proposition.
Let
X
be
a
set
and
let
S
be
any
collection
of
subsets
of
X
such
that
X
=
V.
Vof
∈Sthe
B(��
�)
⊆
R
such
that
�
∈
Q
and
�
=
(�
�
�
�
�
�
�
�
�
)
where
�
�
�
�
�
�
�
∈
Q.
Then
B
is
a
basis
�
�
1 2
1R
Proof. Exercise.
� (exercise).
Let
T
denote
the
be{��
obtained
twoBoperations:
Euclidean
topology
4.8
Example.
Letcollection
X on
=R
{��
��of��all
�}subsets
and let�ofBX=that
{{��can
��� �}�
�� �}}. using
The set
is not a basis of any
�
topology
on
X
since
�
∈
{��
��
�}
∩
{��
��
�},
and
B
does
not
contain
any
subset
Wthat
suchXthat
W
1) Definition.
taking finiteLet
intersections
sets
4.10
X be a set ofand
letinSS;be any collection of subsets of X such
= �V ∈
∈S V .
and
W
⊆
{��
��
�}
∩
{��
��
�}.
4.62)Example.
set Bby=Proposition
{[�� �) | ��
�
is a basis
of a certain
topology
R. the
Wecollection
will call it
The
topology
TThe
defined
4.9∈isR}
called
the topology
generated
by S,onand
S
taking arbitrary
4.7
Example.
Let B = unions
{[�� �] |of��sets
� ∈obtained
R}. The in
set1).
B is a basis of the discrete topology on R (exercise).
thecalled
arrowatopology.
is
subbasis of T.
ThenProposition.
T is a topology
X .a set and let S be any collection of subsets of X such that X = �
4.9
Let Xonbe
V ∈S V .
4.8
Example.
Let
X
=
{��
��
��
�}
and
let
B
=
{{��
��
�}�
{��
��
�}}.
The
set
B
is
not
a
basis
of any
Let TExample.
denote theIf collection
X that
can
two operations:
4.11
X = {�� ��of��all
�} subsets
and S =of{{��
�� �}�
{��be��obtained
�}}Rthenusing
the topology
generated by S is
topology
on
X
since
�
∈
{��
��
�}
∩
{��
��
�},
and
B
does
not
contain
any
subset
W
such
that
�
∈W
Proof.
Exercise.
T=
{{��
�� �}� {�� intersections
�� �}� {���}� {�� �� �� �}� ?}.
�
and1)Wtaking
⊆ {��finite
�� �} ∩ {�� �� �}. of sets in �S;
�
2) taking arbitrary unions of sets obtained in 1).
4.10
Definition. Let
be any collection
subsets
of Xa basis
such that
= � V ∈S V
4.12 Proposition.
LetX(Xbe� TaX ),set(Yand
� TY )let
beStopological
spaces, of
and
let B be
(or aXsubbasis)
of.
4.9
Proposition.
Let
Xon
be
a set
and
let
SThe
be
any
collection
ofofsubsets
of Xbytopology
such
that
XR
=(exercise).
VS.
4.7
Example.
Let
B
=
{[��
�]
|
��
�
∈
R}.
set
B
is
a
basis
the
discrete
on
−1
V
∈S
Then
T
is
a
topology
X
.
The
topology
T
defined
by
Proposition
4.9
is
called
the
topology
generated
S,
and
the
collection
TY . A function � : X → Y is continuous iff � (V ) ∈ TX for every V ∈ B.
Letcalled
T denote
the collection
is
a subbasis
of T. of all subsets of X that can be obtained using two operations:
4.8
Let Xintersections
= {�� �� �� �}
and in
letS;B = {{�� �� �}� {�� �� �}}. The set B is not a basis of any
1)Example.
taking
finite
of sets
Proof.
Exercise.
topology
on
X
since
�
∈
{��
��
�}
∩
{��
and
B
not�}}
contain
subset Wgenerated
such that by
� ∈SW
4.11
If X = unions
{�� �� of sets
and
S��=�},
{{��
�}�does
{�� ��
then any
the topology
is
2) Example.
taking arbitrary
obtained
in��1).
�
and
W
⊆
{��
��
�}
∩
{��
��
�}.
T
=
{{��
��
�}�
{��
��
�}�
{���}�
{��
��
��
�}�
?}.
4.10
Let XonbeX .a set and let S be any collection of subsets of X such that X = V ∈S V .
Then Definition.
T is a topology
The topology T defined by Proposition 4.9 is called the topology generated by S, and the collection
S
�
4.9
Proposition.
Let
X
be
a
set
and
let
S
be
any
collection
of
subsets
of
X
such
that
X
=
V
.
4.12
Proposition.
Let
(X
�
T
),
(Y
�
T
)
be
topological
spaces,
and
let
B
be
a
basis
(or
a
subbasis)
of
V
∈S
X
Y
is called a subbasis of T.
−1
Let
the
all subsets
be every
obtained
Proof.
Exercise.
TY . TA denote
function
� : collection
X → Y is of
continuous
iff of
� X(Vthat
) ∈ can
TX for
V ∈using
B. two operations:
1) Example.
taking finite
of and
sets Sin=S;{{�� �� �}� {�� �� �}} then the topology generated�by S is
4.11
If Xintersections
= {�� �� �� �}
4.10
Definition.
Let
X be
a set{��
and
let�}�
S be
collection of subsets of X such that X = V ∈S V .
T=
�� �}�
{��
�� �}�
{���}�
��obtained
��
?}.any
2){{��
taking
arbitrary
unions
of sets
in 1).
The topology T defined by Proposition 4.9 is called the topology generated by S, and the collection S
Then
T isaasubbasis
topologyofonT.X .
is
called
4.12
Proposition.
Let (X � T ), (Y � T ) be topological spaces, and let B be a basis (or a subbasis) of
X
Y
TY . A function � : X → Y is continuous iff � −1 (V ) ∈ TX for every V ∈ B.
4.11
Proof.Example.
Exercise.If X = {�� �� �� �} and S = {{�� �� �}� {�� �� �}} then the topology generated by S is
T = {{�� �� �}� {�� �� �}� {���}� {�� �� �� �}� ?}.
�
4.10 Definition. Let X be a set and let S be any collection of subsets of X such that X = V ∈S V .
4.12topology
Proposition.
Let (Xby� TProposition
andgenerated
let B be abybasis
(orthe
a subbasis)
The
T defined
4.9topological
is called thespaces,
topology
S, and
collection of
S
X ), (Y � TY ) be
−1 (V ) ∈ T for every V ∈ B.
T
.
A
function
�
:
X
→
Y
is
continuous
iff
�
isY called a subbasis of T.
X
4.11 Example. If X = {�� �� �� �} and S = {{�� �� �}� {�� �� �}} then the topology generated by S is
T = {{�� �� �}� {�� �� �}� {���}� {�� �� �� �}� ?}.
4. Basis, Subbasis, Subspace
27
4.12 Proposition. Let (X � TX ), (Y � TY ) be topological spaces, and let B be a basis (or a subbasis) of
TY . A function � : X → Y is continuous iff � −1 (V ) ∈ TX for every V ∈ B.
Proof. Exercise.
4.4 Definition. Let B be a basis on a set X and let T be the topology defined as in Proposition 4.3.
In such case we will say that B is a basis of the topology T and that T is the topology defined by the
31
basis B.
4.5 Example. Consider R� with the Euclidean metric �. Let B be the collection of all open balls
�
4. Basis, Subbasis, Subspace
28
Proof. Exercise.
4.13 Definition. Let (X � T) be a topological space and let Y ⊆ X . The collection
TY = {Y ∩ U | U ∈ T}
is a topology on Y called the subspace topology. We say that (Y � TY ) is a subspace of the topological
space (X � T).
X
Y
4.14 Example. The unit circle S 1 is defined by
U ∩Y
U is open in X
U
U ∩ Y is open in Y
S 1 := {(�1 � �2 ) ∈ R2 | �12 + �22 = 1}
The circle S 1 is a topological space considered as a subspace of R2 .
open in R2
open in S 1
In general the �-dimensional sphere S � is defined by
2
= 1}
S � := {(�1 � � � � � ��+1 ) ∈ R�+1 | �12 + · · · + ��+1
It is a topological space considered as a subspace of R�+1 .
4.15 Example. Consider Z as a subspace of R. The subspace topology on Z is the same as the discrete
topology (exercise).
32
TY = {Y ∩ U | U ∈ T}
is a topology on Y called the subspace topology. We say that (Y � TY ) is a subspace of the topological
U ∩ Y is open in Y
space (X � T).
4.14 Example. The unit circle S 1 is defined by
1
X
U is open in X
+ �22 = 1}
S := {(�1 � �2 ) ∈ R2 | �12 U
Y
U ∩Y
The circle S 1 is a topological space considered as a subspace of R2 .
4.14 Example. The unit circle S 1 is defined by
open in R2
U ∩ Y is open in Y
open in S 1
S 1 := {(�1 � �2 ) ∈ R2 | �12 + �22 = 1}
The circle S 1 is a topological space considered as a subspace of R2 .
In general the �-dimensional sphere S � is defined by
open in R2
2
��+1
=S
1}1
S � := {(�1 � � � � � ��+1 ) ∈ R�+1 | �12 + · · · + open
in
It is a topological space considered as a subspace of R�+1 .
4.15 Example. Consider Z as a subspace of R. The subspace topology on Z is the same as the discrete
topology (exercise).
In general the �-dimensional sphere S � is defined by
2
= 1}
S � := {(�1 � � � � � ��+1 ) ∈ R�+1 | �12 + · · · + ��+1
It is a topological space considered as a subspace of R�+1 .
4.15 Example. Consider Z as a subspace of R. The subspace topology on Z is the same as the discrete
topology (exercise).
33
4. Basis, Subbasis, Subspace
29
4.16 Proposition. Let X be a topological space and let Y be its subspace.
1) The inclusion map � : Y → X is a continuous function.
2) If Z is a topological space then a function � : Z → Y is continuous iff the composition �� : Z → X is
continuous.
Proof. Exercise.
4.17 Proposition. Let X be a topological space and let Y be its subspace. If B is a basis (or a
subbasis) of X then the set BY = {U ∩ Y | U ∈ B} is a basis (resp. a subbasis) of Y .
4.
Basis,
Subbasis, Subspace
Proof.
Exercise.
29
4.16 Proposition. Let X be a topological space and let Y be its subspace.
1)Exercises
The inclusion map
� : Y → X is a4
continuous function.
to Chapter
2) If Z is a topological space then a function � : Z → Y is continuous iff the composition �� : Z → X is
continuous.
E4.1 Exercise. Prove Proposition 4.3
E4.2 Exercise. Verify the statement of Example 4.5.
Proof. Exercise.
E4.3 Exercise. Verify the statement of Example 4.7.
4.17 Proposition. Let X be a topological space and let Y be its subspace. If B is a basis (or a
E4.4 Exercise.
Prove
4.9.∩ Y | U ∈ B} is a basis (resp. a subbasis) of Y .
subbasis)
of X then
theProposition
set BY = {U
E4.5 Exercise. Prove Proposition 4.12.
Proof. Exercise.
E4.6 Exercise. Consider the interval [0� 1] as a subspace of R. Determine which of the following sets
are open in [0� 1]. Justify your answers.
�
�
�
�
�
�
�
�
b) 12 � 1
c) 13 � 23
d) 13 � 23
a) 12 � 1
Exercises to Chapter 4
E4.7 Exercise. Verify the statement of Example 4.15.
E4.1 Exercise. Prove Proposition 4.16.
4.3
E4.8
E4.2 Exercise. Prove
Verify Proposition
the statement
of Example 4.5.
E4.9
4.17.
E4.3 Exercise.
thethat
statement
of Example
We say
a topological
space4.7.
X satisfies the 2 nd countability axiom (or that it is
E4.10
Exercise.Verify
second countable) if the topology on X has a countable basis. The basis of the Euclidean topology on
E4.4 Exercise. Prove Proposition 4.9.
R� described in Example 4.5 is countable, which shows that R� satisfies this axiom.
E4.5 Exercise. Prove Proposition 4.12.
Show that R with the arrow topology is not second countable. (Hint: Assume by contradiction that
B
= {V
} is a countable
basis of
arrow
topology.ofLet
Take α0of∈the
Rr{α
� � }.
� = inf V� . which
1 � V2 � � � � Consider
1 � α2 � �sets
the interval
[0�the
1] as
a subspace
R. αDetermine
following
E4.6
Exercise.
Show
that
the
set
[α
�
α
+
1)
is
not
a
union
of
sets
from
B).
0 0 your answers.
are open in [0� 1]. Justify
�
�
�
�
�
�
�1 �
b) 12 � 1
c) 13 � 23
d) 13 � 23
a) 2 � 1
E4.7 Exercise. Verify the statement of Example 4.15.
E4.8 Exercise. Prove Proposition 4.16.
E4.9 Exercise. Prove Proposition 4.17.
is
E4.10 Exercise. We say that a topological space X satisfies the 2 nd countability axiom (or that it 34
second countable) if the topology on X has a countable basis. The basis of the Euclidean topology on
R� described in Example 4.5 is countable, which shows that R� satisfies this axiom.