MA4266 Topology

1
MA4266 Topology
Lecture 12

• Wayne Lawton
• Department of Mathematics
• S17-08-17, 65162749 matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml/
• http//arxiv.org/find/math/1/auLawton_W/0/1/0/al
  l/0/1

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Local Compactness
Definition A space
is locally compact at a point
if there exists an open set
which contains
and such that
is compact.
A space is locally
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)
3
Examples
Example 6.4.1
is locally compact.
(a)
(b)
is not locally compact.
Supplemental Example
Definition An operator
(this means a function that is continuous and
linear)
is called compact if it maps any bounded set
onto a relatively compact set, this means that
is compact (equivalent to
totally bounded)
http//en.wikipedia.org/wiki/Compact_operator
Question Is
a compact operator ?
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One-Point Compactification
be a topological space and
Definition Let
called the point at infinity, be an object not in

Let
and
a topology on
Question Why is
Theorem 6.18 (proofs given on page 183)
(a)
is compact.
(b)
is a subspace of
(c)
is Hausdorff iff
is Hausdorff locally compact
(d)
is dense in
iff
is not compact.
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Stereographic Projection
Question What is the formula that maps
onto
?
Question Why is
homeomorphic to
?
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The Cantor Set
Definition The Cantor (ternary) set is
are defined by
where
is obtained from
by removing the middle
open third (interval) from each of the
closed intervals whose union equals
Question What is the Lebesgue measure of
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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.ht
ml
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
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Fat Cantor Sets

• SmithVolterraCantor 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 SmithVolterraCantor set is named
  after the
• mathematicians Henry Smith, Vito Volterra and
  Georg Cantor.

The SmithVolterraCantor 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/FileSmith-Volterra-C
antor_set.svg
http//www.macalester.edu/bressoud/talks/Alleghen
yCollege/Wrestling.pdf
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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
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Supplementary Materials
be a compact metric space and
Definition Let
be the metric space of real-valued
continuous functions on
with the following metric
Definition A subset
is equicontinuous
there exists
such that
if for every
and
uniformly bounded if
is bounded.
Theorem (ArzelàAscoli)
is relatively compact iff it is
uniformly bounded and equicontinuous.
http//en.wikipedia.org/wiki/ArzelC3A0E28093A
scoli_theorem
http//www.mth.msu.edu/shapiro/Pubvit/Downloads/A
rzNotes/ArzNotes.pdf