Transcript MA4266_Lect12 - Department of Mathematics

MA4266 Topology
Lecture 12
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
Local Compactness
X is locally compact at a point
p  X if there exists an open set U which contains
p and such that U is compact. A space is locally
Definition: A space
compact if it is locally compact at each of its points.
Question Is local compactness a topological property?
Question Is local compactness a local property ?
(compare with local connectedness and local path
connectedness to see the apparent difference)
Examples
Example 6.4.1
n
(a) R is locally compact.
(b)  2 ( N ) is not locally compact.
Supplemental Example
Definition An operator
L :  (N )   (N )
2
2
(this means a function that is continuous and linear)
is called compact if it maps any bounded set
X
onto a relatively compact set, this means that
is compact (equivalent to
L(X )
L(X )
totally bounded)
http://en.wikipedia.org/wiki/Compact_operator
Question Is
L({xk })  { xk }
1
k
a compact operator ?
One-Point Compactification
Definition Let ( X , T ) be a topological space and
, called the point at infinity, be an object not in X .
Let X  X {} and T  T { X }



{O  X  : X  \ O is a closedcompactsubset of X }
Question Why is
T
a topology on
X ?
Theorem 6.18: (proofs given on page 183)
(a)
(b)
( X  ,T ) is compact.
( X,T) is a subspace of ( X  ,T )
(c) X  is Hausdorff iff X is Hausdorff & locally compact
(d)
X
is dense in
X  iff X
is not compact.
Stereographic Projection
Question What is the formula that maps
n
S \ {northpole}
Question Why is
S
onto
n
R
n
?
homeomorphic to
n
( R ) ?
The Cantor Set
Definition: The Cantor (ternary) set is
where
Fk  [0,1], k  1,2,3,...

C   k 1 Fk
are defined by
F1  [0, ] [ ,1]
1
3
2
3
F2  [0, 19 ] [ 92 , 13 ] [ 23 , 79 ] [ 89 ,1]

Fk 1
is obtained from Fk by removing the middle
open third (interval) from each of the 2 k
closed intervals whose union equals Fk
Question What is the Lebesgue measure of
C?
Properties of Cantor Sets
Definition: A closed subset A of a topological space X
is called perfect if every point of A is a limit point of A.
X is called scattered if it contains no perfect subsets.
http://planetmath.org/encyclopedia/ScatteredSet.html
Theorem 6.19: The Cantor set is a compact, perfect,
totally disconnected metric space.
Theorem Any space with these four properties is
homeomorphic to a Cantor set.
Remark There are topological Cantor sets, called
fat Cantor sets, that have positive Lebesgue measure
Fat Cantor Sets
Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an
example of a set of points on the real line R that is nowhere
dense (in particular it contains no intervals), yet has positive
measure. The Smith–Volterra–Cantor set is named after the
mathematicians Henry Smith, Vito Volterra and Georg Cantor.
The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].
The process begins by removing the middle 1/4 from the interval [0, 1] to obtain
The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remaining
intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get
http://en.wikipedia.org/wiki/File:Smith-Volterra-Cantor_set.svg
http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf
Assignment 12
Read pages 181-190
Prepare to solve during Tutorial Thursday 11 March
Exercise 6.4 problems 9, 12
Exercise 6.5 problems 3, 6, 9
Supplementary Materials
Definition: Let
(X ,d)
(C( X ), dmax )
be a compact metric space and
be the metric space of real-valued
X with the following metric:
dmax ( f , g )  maxxX | f ( x)  g ( x) |
continuous functions on
S  C (X ) is equicontinuous
if for every   0 there exists   0 such that
Definition A subset
d ( x, y)    dmax ( f ( x), f ( y))   ,
f  S , x, y  X and
uniformly bounded if { f ( x) : f  S , x  X } is bounded.
Theorem (Arzelà–Ascoli): S is relatively compact iff it is
uniformly bounded and equicontinuous.
http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem
http://www.mth.msu.edu/~shapiro/Pubvit/Downloads/ArzNotes/ArzNotes.pdf