1. Department of Mathematics
MA4266 Topology
Lecture 16
Wayne Lawton
Department of Mathematics
S ,

2. 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 & Thomas’s Corkscrew Top., Ostaszewski
points & closed sets jointly separated by CF
Ex. Sorgenfrey plane
pairs of closed sets jointly separated by OS 2

3. Combinations of Separation Properties
Definition A space is
Completely Hausdorff or 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 3

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. 4

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.
Proof Let
be a family of regular spaces,
Therefore
Then
why?
Then
is open, contains
and so
is regular. 5

6. Examples Double Origin Topology (counterexample # 74, [1])
has a local basis
Question Why is
Question Why is
NOT
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. 6

7. Examples Half-Disc Topology (counterexample # 78, [1])
and Example in Croom’s Principles of Topology.
where a local basis at is
Question Why is
Question Why is
NOT 7

8. Normal Spaces Theorem 8.6 Assume that is a space. Then is
(and hence normal) iff for every closed
and
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 8

9. Normal Spaces Theorem 8.9 Every regular Lindelöf space is normal.
Proof Let
be disjoint closed sets.
First, use
regularity to construct an open cover of
by sets
whose closures are disjoint with
likewise for
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 9

10. 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 10

11. Examples Sorgenfrey Plane (counterexample # 84, [1])
and Example in Croom’s 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 ? 11

12. Examples Niemytzki’s Tangent Disc Top. (counterexample # 82,[1])
and Ex. 8.3, Q6, p. 242 Croom’s Principles of Topology.
where a local basis at
Question Why is
Question Why is
separable ?
Question Why is
NOT normal ? 12

13. Separation by Continuous Functions
Definition Separation by continuous functions.
and Ex , Q6, p. 243 Croom’s 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. 13

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. 14

15. 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
and therefore
is constant on 15

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 16

17. 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 Urysohn’s 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 17

18. 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 18