MA4266 Topology Wayne Lawton Department of Mathematics S , Lecture 12. Tuesday 9 March 2010

Local Compactness Definition: A spaceis locally compact at a point if there exists an open setwhich contains and such thatis compact. 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 A space is locally connectedness to see the apparent difference)

Examples Example is locally compact.(a) Supplemental Example Definition An operator (this means a function that is continuous and linear) is called compact if it maps any bounded set (b)is not locally compact. onto a relatively compact set, this means that is compact (equivalent tototally bounded) Question Isa compact operator ?

One-Point Compactification be a topological space andDefinition Let called the point at infinity, be an object not in Let Theorem 6.18: (proofs given on page 183) a topology on and Question Why is (a) (d) (c) (b) is compact. is a subspace of is Hausdorff iffis Hausdorff & locally compact is dense iniffis not compact.

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

The Cantor Set are defined bywhere Definition: The Cantor (ternary) set is is obtained fromby removing the middle open third (interval) from each of the closed intervals whose union equals Question What is the Lebesgue measure of

Properties of Cantor Sets is called perfect if every point of A is a limit point of A. Definition: A closed subset A of a topological space X Theorem 6.19: The Cantor set is a compact, perfect, Theorem Any space with these four properties is homeomorphic to a Cantor set. X is called scattered if it contains no perfect subsets. totally disconnected metric space. Remark There are topological Cantor sets, called fat Cantor sets, that have positive Lebesgue measure

Fat Cantor Sets Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowherereal linenowhere densedense (in particular it contains no intervals), yet has positiveintervals measuremeasure. The Smith–Volterra–Cantor set is named after the mathematiciansmathematicians Henry Smith, Vito Volterra and Georg Cantor.Henry SmithVito VolterraGeorg Cantor The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].unit interval 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 1/2 2n from the middle of each of the 2 n−1 remaining intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get

Assignment 12 Read pages Prepare to solve during Tutorial Thursday 11 March Exercise 6.4 problems 9, 12 Exercise 6.5 problems 3, 6, 9

Definition: Let Supplementary Materials Definition A subset there exists Theorem (Arzelà–Ascoli): such that uniformly bounded if be a compact metric space and be the metric space of real-valued continuous functions onwith the following metric: is equicontinuous if for every is bounded. and is relatively compact iff it is uniformly bounded and equicontinuous.