MA4266 Topology

1
MA4266 Topology
Lecture 16

• 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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Separation Properties
or axioms, specify the degree by which points
and/or closed sets
can be separated by open sets continuous
functions
Kolmogorov space
Ex. Sierpinski space
1 point from a pair separated from the other by OS
Frechet space
Ex. Finite Comp. Top. on Z
each point from a pair separated from the other
by OS
Hausdorf space
Ex. Double Origin Top. on
pairs of points jointly separated by OS
completely Hausdorf space (called Urysohn in
book)
PP sep. by CN Ex. Half-Disc Top. on
points closed sets jointly separated by OS
Ex. Tychonov Hewitt Thomass Corkscrew Top.,
Ostaszewski
points closed sets jointly separated by CF
Ex. Sorgenfrey plane
pairs of closed sets jointly separated by OS
3
Combinations of Separation Properties
Completely Hausdorff or
Definition A space
is
if
Regular if it is
and
Completely Regular or
if it is
and
Normal if it is
and
Theorem 8.1
finite subsets are closed.
Metrizable ?Normal ? Completely Regular ? Regular
?
?
?
?
Theorem 8.2 Products of
spaces are
4
Regular Spaces
Theorem 8.3 Assume that
is a
space.
Then
is
(and therefore regular) if and only if for every
and open
there exists open
with
Proof If
is regular and
then
is closed and
hence there exist
disjoint open
Hence
Hence
(why?)
so
Conversely, if the latter condition holds and
is a closed
set with
Then there exists open
with
(why?) so
and
are disjoint open
sets containing
and
respectively.
5
Regular Spaces
Theorem 8.4 Assume that
is a
space.
Then
is
if and only if for every
there exists open
with
Proof page 235.
Theorem 8.5 The product of regular spaces is
regular.
be a family of regular spaces,
Proof Let
Therefore
why?
Then
Then
is open, contains
and
so
is regular.
6
Examples
Double Origin Topology (counterexample 74, 1)
has a local basis
Question Why is
NOT
Question Why is
Question Is
2nd countable ? path connected ?
Question Is
regular ? locally compact ?
1 Counterexamples in Topology by Lynn Arthur
Steen and J. Arthur Seebach, Jr., Dover, New
York, 1970.
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Examples
Half-Disc Topology (counterexample 78, 1)
and Example 8.2.1 in Crooms Principles of
Topology.
where a local basis at
is
Question Why is
NOT
Question Why is
8
Normal Spaces
Theorem 8.6 Assume that
is a
space.
Then
is
and
(and hence normal) iff for every closed
open
there exists open
with
Theorem 8.7 Assume that
is a
space.
Then
is
iff for every pair of disjoint closed sets
there exist open sets
with
Theorem 8.8 Every compact Hausdorff space is
normal.
Proof Corollary to Theorem 6.5, pages 165-166.
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Normal Spaces
Theorem 8.9 Every regular Lindelöf space is
normal.
Proof Let
First, use
be disjoint closed sets.
regularity to construct an open cover of
by sets
likewise for
whose closures are disjoint with
Second, use the Lindelöf property to obtain
countable
subcovers
of
and
of
Third, construct
and observe that
Fourth, construct
and
observe they are open sets and
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Normal Spaces
Why ?
Corollary Every 2nd countable regular space is
normal.
Definition For a set
Theorem 8.10 If
is a separable normal space and
is a subset with
then
has a limit point.
Proof Assume that such a set
has no limit point. Then
for every
the sets
and
are closed
so there exist disjoint open
and
Let
Be a countable dense subset and construct a
function
by
Since
is 1-to-1 (see p. 239)
But
Theorem 8.11 Every metric space is normal. Ex 3.2
p.69
11
Examples
Sorgenfrey Plane (counterexample 84, 1)
and Example 8.3.1 in Crooms Principles of
Topology.
Question Why is
regular ?
Lindelöf ?
Question Why is
normal ?
Question Why is
regular, separable ?
Question Why is
Let
Question What is the subspace topology on
Question What are the limit points of
Question Why is
NOT normal ?
12
Examples
Niemytzkis Tangent Disc Top. (counterexample
82,1)
and Ex. 8.3, Q6, p. 242 Crooms Principles of
Topology.
where a local basis at
Question Why is
Question Why is
separable ?
NOT normal ?
Question Why is
13
Separation by Continuous Functions
Definition Separation by continuous functions.
and Ex. 8.4.1, Q6, p. 243 Crooms Principles of
Top.
Theorem 8.12 Let
be a
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.
14
Examples
Definition Funny Line
where
is open iff
is finite.
(a one-point compactification of an uncountable
set)
Definition A subset S of a topological space X is
a
set (gee-delta) if it is the intersection of a
countable collection of open sets, and a
set (eff-sigma) if it is the union of a
countable collection of closed sets.
Theorem If
is a topological space and
is
continuous then
is a
set for every
Proof
Corollary Every continuous
equals
except at a countable set of points.
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Examples
Thomas Plank (counterexample 93, 1)
where
Theorem If
is continuous then
is constant
except at a countable set of points.
Proof On each set
the function
is constant
countable.
on a set
where
is constant on each
where
is constant on
and therefore
16
Examples
Thomas Corkscrew (counterexample 94, 1)
where the local bases for points in
is the same as for the product topology,
and local bases
for other points are
for
Theorem
is regular but NOT completely regular
since every continuous
satisfies
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Separation by Continuous Functions
Lemma 1. Dyadic numbers are dense in
Lemma 2. Let
be a space and
If for every
(a)
and
(b)
then the function
defined by
is continuous.
Theorem 8.13 Urysohns Lemma Let
be a
space.
then
is normal iff for all disjoint closed
there exists a continuous
with
Theorem 8.14 Tietze Extension Theorem Let
be a
normal space,
and
continuous.
Then
has a continuous extension
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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