MA4266 Topology

1. Lecture 12. Tuesday 9 March 2010 MA4266 Topology 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

2. 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 (a) is locally compact. (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 ?

4. One-Point Compactification Definition Let be a topological space and called the point at infinity, be an object not in Let and Question Why is a topology on 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.

5. Stereographic Projection Question What is the formula that maps onto ? Question Why is homeomorphic to ?

6. The Cantor Set Definition: The Cantor (ternary) set is where are defined by is obtained from by removing the middle open third (interval) from each of the closed intervals whose union equals QuestionWhat is the Lebesgue measure of

7. 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.html 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

8. Fat Cantor Sets Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real lineR that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematiciansHenry Smith, Vito Volterra and Georg Cantor. The Smith–Volterra–Cantor 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/File:Smith-Volterra-Cantor_set.svg http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf

9. 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

10. 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 if for every there exists such that and uniformly bounded if is bounded. Theorem (Arzelà–Ascoli): is relatively compact iff it is uniformly bounded and equicontinuous. http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem http://www.mth.msu.edu/~shapiro/Pubvit/Downloads/ArzNotes/ArzNotes.pdf