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MA4266 Topology
Lecture 16
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
Separation Properties
or axioms, specify the degree by which points and/or closed sets
can be separated by open sets & continuous functions
T0 Kolmogorov space
Ex. Sierpinski space
1 point from a pair separated from the other by OS
T1 Frechet space Ex. Finite Comp. Top. on Z
each point from a pair separated from the other by OS
~
2
T2 Hausdorf space Ex. Double Origin Top. on R  { 0 }
pairs of points jointly separated by OS
T2 1 completely Hausdorf space (called Urysohn in book)
2
PP sep. by CN
Ex. Half-Disc Top. on
R  [0,  )
T3 points & closed sets jointly separated by OS
Ex. Tychonov & Hewitt & Thomas’s Corkscrew Top., Ostaszewski
T3 1 points & closed sets jointly separated by CF
2
T4
Ex. Sorgenfrey plane
pairs of closed sets jointly separated by OS
Combinations of Separation Properties
X is Completely Hausdorff or T2 12
if a, b  X , a  b   open U  a,V  b,U  V  .
Regular if it is T1 and T3 , Completely Regular or T3 1
2
if it is T1 and a  X , a  closed C  X 
Definition A space
 continuous f:X  R, f (a)  0, f (C )  1,
Normal if it is
Theorem 8.1
T1 and T4 .
T1  finite subsets are closed.
Metrizable Normal  Completely Regular  Regular

T2 1  T2  T1  T0
2
Theorem 8.2 Products of
Ti spaces are Ti , i  0,1, 2.
Regular Spaces
X is a T1 space. Then X
is T3 (and therefore regular) if and only if for every a  X
and open U  a there exists open W  a with W  U .
Proof If X is regular and a  U open  X then
C  X \U is closed and a  C hence there exist
disjoint open W  a and V  C. Hence W  X \V .
Theorem 8.3 Assume that
W  X \ V (why?) so W  X \ C  U .
Conversely, if the latter condition holds and C is a closed
set with a  C. Then there exists open W  a with
W  X \ C (why?) so W and X \ W are disjoint open
sets containing a and C respectively.
Hence
Regular Spaces
X is a T1 space. Then X
is T3 if and only if for every a  X , a  C closed,
there exists open U , V with a U , C  V ,U  V  .
Theorem 8.4 Assume that
Proof page 235.
Theorem 8.5 The product of regular spaces is regular.
Proof Let { X 
:   A} be a family of regular spaces,
X   A X  , a U open  X. Therefore
n
1
U  i open  X , i  1,..., n  a   i 1 p (U  )  U .
Then  p (a) V open  V  U , i  1,..., n why?
n
1
V

p
(
V
)
Then
is open, contains a and



i 1
n
n
1
V   i 1 p (V )   i 1 p1 (U  )  U , so X is regular.
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Examples
Double Origin Topology (counterexample # 74, [1])
( X , ), X  ( R \ {0})  {0 } { 0 }
2
  ( R \ {0})usual  0  0
 0  has a local basis
2
2
Vn,   { ( x, y) : x  y  n12 ,  y  0} {0 }, n  N
Question Why is ( X , ) T2 ?
Question Why is ( X , ) NOT T2 1 ?
2
Question Is ( X , ) 2nd countable ? path connected ?
Question Is ( X , ) regular ? locally compact ?
2
[1] Counterexamples in Topology by Lynn Arthur Steen
and J. Arthur Seebach, Jr., Dover, New York, 1970.
Examples
Half-Disc Topology (counterexample # 78, [1])
and Example 8.2.1 in Croom’s Principles of Topology.
( X , ), X  R  (0, )  R  {0}
  ( R  (0, )) usual  ( R {0})half disk
where a local basis at
(a,0)  R  {0} is
Vn  { ( x, y) : ( x  a) 2  y 2  n12 , y  0} {( a,0)}, n  N
Question Why is
( X , ) T2 1 ?
2
Question Why is
( X , ) NOT T3 ?
Normal Spaces
X is a T1 space. Then X
is T4 (and hence normal) iff for every closed A  X and
open U  A there exists open W  A with W  U .
Theorem 8.7 Assume that X is a T1 space. Then X
is T iff for every pair of disjoint closed sets A, B  X
4
there exist open sets U  A, V  B with U V   .
Theorem 8.6 Assume that
Theorem 8.8 Every compact Hausdorff space is normal.
Proof Corollary to Theorem 6.5, pages 165-166.
Normal Spaces
Theorem 8.9 Every regular Lindelöf space is normal.
A, B be disjoint closed sets. First, use
regularity to construct an open cover of A by sets
whose closures are disjoint with A, likewise for B.
Proof Let
Second, use the Lindelöf property to obtain countable
subcovers
U1 ,U 2 ,... of A
and
n
V1 ,V2 ,...
B.
of
n
Third, construct U  U n \  i 1 Vn , V  Vn \  i 1 U n
'
n
and observe that
'
n
U  V   , for all m, n  N .
'
n
Fourth, construct U 
'
m


'
U
,
V

V
 n1
 n1 n and
'
n
observe they are open sets and A  U , B  V .
Normal Spaces
Why ?
Corollary Every 2nd countable regular space is normal.
Definition For a set A, card A  c  A is equipotent with [0,1]
X is a separable normal space and E
card E  c, then E has a limit point.
Proof Assume that such a set E has no limit point. Then
for every Y  E the sets Y and E \ Y are closed
so there exist disjoint open UY  Y and VY  E \ Y .
Let D Be a countable dense subset and construct a
Theorem 8.10 If
is a subset with
function h : P( E )  P( D) by h(Y )  UY  D, Y  P( E).
Since h is 1-to-1 (see p. 239) card P( E )  card P( D).
But card P( D)  card R  card P( R)  card P( E ).
Theorem 8.11 Every metric space is normal. Ex 3.2 p.69
Examples
Sorgenfrey Plane (counterexample # 84, [1])
and Example 8.3.1 in Croom’s Principles of Topology.
( R  R,  )   half  open interval topology
Question Why is ( R, ) regular ?
Question Why is ( R, ) Lindelöf ?
Question Why is ( R, ) normal ?
Question Why is ( R  R,   ) regular, separable ?
Let E  {( x, x) : x  R}  R  R
Question What is the subspace topology on E ?
Question What are the limit points of E ?
Question Why is ( R  R,   ) NOT normal ?
Examples
Niemytzki’s Tangent Disc Top. (counterexample # 82,[1])
and Ex. 8.3, Q6, p. 242 Croom’s Principles of Topology.
( X , ), X  R  (0, )  R  {0}
  ( R  (0, )) usual  ( R  {0}) tangentdisk
where a local basis at (a,0)  R  {0}
Vn  { ( x, y) : ( x  a)  ( y  )  n12 } {( x,0)}, n  N
2
1 2
n
( X , ) T3 1 ?
2
Question Why is ( X , ) separable ?
Question Why is ( X , ) NOT normal ?
Question Why is
Separation by Continuous Functions
Definition Separation by continuous functions.
and Ex. 8.4.1, Q6, p. 243 Croom’s Principles of Top.
Theorem 8.12 Let
X be a T1 space.
(a) If points a and b can be separated by a continuous
function then they can be separated by open sets.
(b) If each point x and closed set C not containing a
can be separated by continuous functions then
they can be separated by open sets.
(c) If disjoint closed sets A and B can be separated
continuous functions then they can be separated
by open sets.
Examples
Definition Funny Line :
( X 1 , 1 ), X 1  R,  1  ( R \ {0})discrete  {0}
where U  0 is open iff R \ U is finite.
(a one-point compactification of an uncountable set)
Definition A subset S of a topological space X is a
G  set (gee-delta) if it is the intersection of a
countable collection of open sets, and a
F  set (eff-sigma) if it is the union of a
countable collection of closed sets.
Theorem If X is a topological space and f : X  R is
1
f
(a) is a G  set for every a X .
continuous then
1
Proof f (a)  f
1


i 1


B(a, )   i 1 f 1 ( B(a, 1i )).
1
i
Corollary Every continuous f : X1  R equals f (0)
except at a countable set of points.
Examples
Thomas’ Plank (counterexample # 93, [1])
( X , ), X  X1  X 2 \ {( 0,0)}, prod top
1
(
X
,

),
X

{
where
2
2
2
k : k  Z \ {0}}  {0}
 2  usual subspace topology of R
Theorem If f : X  R is continuous then f is constant
except at a countable set of points.
1
f
R

{
}
Proof On each set
the
function
is constant
k
1
countable.
R

{
}
R
\
R
on a set k
where
k
k
1
L   kZ \{ 0} Rk
L

{
f is constant on each
k } where
and therefore
f is constant on ( L \ {0})  {0}.
Examples
Thomas’ Corkscrew (counterexample # 94, [1])
(Y , ), Y  X  Z  {a }  {a }


where the local bases for points in
Y \ {( x,0, m) : x  0, m  Z }
is the same as for the product topology, and local bases
for other points are
Bn ( x,0, m)  {( x,0, m)}  { ( x, 1k , m) : k  n }  { ( x, 1k , m  1) : k  n }
Bn (a  )  {a  }   m  n X  {m}, Bn (a  )  {a  }   m   n X  {m},
n N.
Theorem (Y , )
for
is regular but NOT completely regular
since every continuous f : Y  R satisfies f (a  )  f (a  ).
Separation by Continuous Functions
Lemma 1. Dyadic numbers are dense in R.
Lemma 2. Let X be a space and D dense  [0, ).
t  D  open U t  X 
(a) t1  t2  U t  U t , and (b)  tD U t  X ,
1
2
then the function f : X  R defined by
f ( x)  glb { t  D : x U t }, x  X is continuous.
Theorem 8.13 Urysohn’s Lemma Let X be a T1 space.
then X is normal iff for all disjoint closed A, B  X ,
If for every
there exists a continuous f : X  R with f ( A)  0, f ( B)  1.
Theorem 8.14 Tietze Extension Theorem Let X be a
normal space, A closed  X , and f : A  R continuous.
Then f has a continuous extension F : X  R.
Assignment 16
Read pages 234-237, 237-241, 243-251
Prepare to solve during Tutorial Thursday 8 April
Exercise 8.2 problem 4 (c)
Exercise 8.3 problem 6 (a),(b),(c),(d)
Exercise 8.4 problems 8 (a),(b), 11, 13, 14 (a),(b) 15, 16