1. Department of Mathematics
MA4266 Topology
Lecture 7.
Wayne Lawton
Department of Mathematics
S ,

2. 1st Countable Spaces Theorem Every metric space is first countable
Proof
is a local basis at
Theorem A 2nd countable space 1st countable and separable.
Proof Assume that
is 2nd countable with
a countable basis
Then for every
the set
is a local basis at
For every nonempty
choose
and define the set
Then
is countable and dense
(why?). 2

3. Bases Theorem 4.8 If is a set and then satisfies:
is a basis for a topology on
iff
(a)
(b)
Proof Follows since 3

4. Bases Theorem 4.9 Bases and are equivalent iff (a) and (b) Remark Let
be the topology generated by
and let
be the topology generated by
Then
condition (a) is equivalent to
(or
is weaker than or
is stronger than
and
condition (b) is equivalent to
See 7.3 Comparison of Topologies on pages 4

5. Subbases Exercise 4.3 Problem 10. If is a set and then the family
of finite
satisfies
intersections of members of
is a basis for a topology on
Then
is called a subbasis for
Example Consider
where
is the usual
topology whose members consists of unions of
open balls.
Then the following set is a subbasis for
Proof
is equivalent to the basis for the usual topology that
consists of open balls
as shown by 5

6. Continuity topological spaces, Definition is continuous at if
is continuous if
is continuous at
is continuous if
Alternative Definition
then following are equiv.
Theorem 4.11 If
(2)
(1)
is continuous,
(3)
(4)
(5) 6

7. Exotic Topologies for R
Sorgenfrey Line = R with the half-open interval
topology generated by
Question Why is
a basis for some topology?
Question Is
dense in
Question Does this topology have a countable basis?
Question Is [0,1] compact ? Is R connected ?
R with the countable complement topology
Question Does R have a countable dense subset ?
Question Does this topology have a countable basis?
Question Is [0,1] compact ? Is R connected ? 7

8. Subspaces Definition Let be a topological space and is a topology Then
It is called the relative or subspace topology.
The pair
is a topological space and is called
a subspace of
Theorem 4.16 A subset
is closed in
iff
for some
closed in
Proof. If
is closed in
then
for some
Then
is closed in
and
 If
for some
closed in
then 8

9. Subspaces Definition A property of topological spaces that
holds for all subspaces is called hereditary.
Example st and 2nd countability.
Example Separability is not heriditary.
Example The Zariski Topology
For n = 1 it is the finite compliment topology
For n > 1 it is not the finite compliment topology
It is not Hausdorff 9

10. Assignment 7 Review Chapters 1- 4.
Study all Exercises, be prepared to present
solutions during the tutorial Thursday 4 Feb
Be prepared for Test 1 on Friday 5 Feb.