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MA4266 Topology
Lecture 7.
Wayne Lawton
Department of Mathematics
S17-08-17, 65162749 [email protected]
http://www.math.nus.edu.sg/~matwml/
http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1
1st Countable Spaces
Theorem Every metric space is first countable
Proof p  X ,{B( p, 1n ) : n  1,2,3,...}
is a local basis at p.
Theorem A 2nd countable space 1st countable
and separable.
Proof Assume that X is 2nd countable with
a countable basis B. Then for every p  X
the set B p  {O  B : p  O} is a local basis at p.
For every nonempty O  B choose xO  O
and define the set D  {xO : (O  B)  (O   )}.
Then D is countable and dense ( D  X ) (why?).
Bases
Theorem 4.8 If X is a set and B  P( X ) then B
is a basis for a topology T on X iff B satisfies:
(a)
O  X
OB
(b) O1  B, O2  B, x  O3  B  x  O3  O1  O2
O1

O3 x
O2
Proof Follows since T  {

OA
O : A  B}
Bases
Theorem 4.9 Bases B and B  are equivalent iff
(a) O  B, x  O, O  B  x  O  O
and
(b) O  B, x  O, O  B  x  O  O
Remark Let T be the topology generated by B
and let T  be the topology generated by B. Then
condition (a) is equivalent to T  T 
(or T is weaker than T , or T  is stronger than T , and
condition (b) is equivalent to T   T
See 7.3 Comparison of Topologies on pages 211-213
Subbases
Exercise 4.3 Problem 10. If X is a set and S  P( X )
satisfies AS A X then the family B of finite
intersections of members of S is a basis for a topology
T on X . Then S is called a subbasis for T .
2
Example Consider ( R , T ) where T is the usual
topology whose members consists of unions of
open balls. Then the following set is a subbasis for T
S  {( a, b)  R : a, b  R, a  b}  {R  (a, b) : a, b  R, a  b}
Proof B  {( a, b)  (c, d ) : a, b, c, d  R, a  b, c  d }
is equivalent to the basis for the usual topology that
consists of open balls
as shown by
Continuity
Definition ( X , T ), (Y , T ) topological spaces, a  X ,
f : X  Y is continuous at a if
V  T  [( f (a) V )  (U  T [( a U )  ( f (U )  V )] )]
f : X  Y is continuous if
a X [ f : X  Y is continuous at a ]
Alternative Definition f : X  Y is continuous if
V  T  [ U  T [ f (U )  V ] ]
Theorem 4.11 If f : X  Y then following are equiv.
1
(2)
(1) f is continuous,
C closed  f (C ) closed
(3) A  X , f ( A )  f ( A)
(4)  basis B for T   O  B, f 1 (O)  T
(5)  subbasis S for T   O  S , f 1 (O)  T
Exotic Topologies for R
Sorgenfrey Line = R with the half-open interval
topology generated by B  {[a, b) : a  b }.
Question Why is B a basis for some topology?
Question Is Q dense in R
Question Does this topology have a countable basis?
Question Is [0,1] compact ? Is R connected ?
R with the countable complement topology
Question Does R have a countable dense subset ?
Question Does this topology have a countable basis?
Question Is [0,1] compact ? Is R connected ?
Subspaces
Definition Let ( X , T ) be a topological space and
A  X . Then T   {O  A : O  T } is a topology
A. It is called the relative or subspace topology.
The pair ( A, T ) is a topological space and is called
a subspace of ( X , T ).
on
Theorem 4.16 A subset D  A is closed in ( A, T )
D  C  A for some C  X closed in ( X , T ).
Proof. If D  A is closed in ( A, T ) then
A \ D  O  A for some OT . Then C  X \ O
iff
is closed in ( X , T ) and D  A \ ( A \ D)  A \ (O  A)  C  A.
 If D  C  A for some C  X closed in ( X , T )
then A \ D  A \ (C  A)  ( X \ C )  A.
Subspaces
Definition A property of topological spaces that
holds for all subspaces is called hereditary.
Example 4.5.1 1st and 2nd countability.
Example 4.5.2 Separability is not heriditary.
Example 4.5.4 The Zariski Topology
For n = 1 it is the finite compliment topology
For n > 1 it is not the finite compliment topology
It is not Hausdorff
http://en.wikipedia.org/wiki/Oscar_Zariski
http://en.wikipedia.org/wiki/Zariski_topology
Assignment 7
Review Chapters 1- 4.
Study all Exercises, be prepared to present
solutions during the tutorial Thursday 4 Feb
Be prepared for Test 1 on Friday 5 Feb.