MA4266_Lect16 - Department of Mathematics
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Transcript MA4266_Lect16 - Department of Mathematics
MA4266 Topology
Lecture 16. Tuesday 6 April 2010
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
Then
is open, contains a and
(V )
i 1
n
n
1
V i 1 p (V ) i 1 p1 (U ) U , so X is regular.
i
i
i
i
i
i
i
i
i
i
i
i
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
n1
n1 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}) tangentdisk
where a local basis at (a,0) R {0}
Vn { ( x, y) : ( x a) ( y ) } {( x,0)}, n N
2
1 2
n
1
n2
( 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 kZ \{ 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) tD 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