Department of Mathematics MA4266 Topology Lecture 7. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

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

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

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 211-213 4

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

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

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

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

Subspaces Definition A property of topological spaces that holds for all subspaces is called hereditary. Example 4.5.1 1st and 2nd countability. Example 4.5.2 Separability is not heriditary. Example 4.5.4 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 http://en.wikipedia.org/wiki/Oscar_Zariski http://en.wikipedia.org/wiki/Zariski_topology 9

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.