Principles of Mathematics in Operations Research
134 9 Number Systems above by bl//'n >- 1. c) If t >• 1 and n >- f^j, then 6^1 /" x i. d) If w 3 6"' X y, then bw+l/n - ...
9.7 Web material 135 http://en.wikipedia.org/wiki/Real_numbers http://en.wikipedia.org/wiki/Uncountable_set http://eom.springer. ...
10 Basic Topology In this chapter, basic notions in general topology will be defined and the re- lated theorems will be stated. ...
138 10 Basic Topology Example 10.1.4 Is / : R H-» R, f(s) = s^2 bounded? (Exercise!). / : R i-> R, f(s) = arctan(s) — tan_1(s ...
10.1 Metric Spaces 139 P<... 6^ -d,— 'Oinfty • Fig. 10.1. Example 10.1.9 Definition 10.1.10 Let (X,d) be a metric space, p 6 ...
140 10 Basic Topology Fig. 10.3. Example 10.1.13 Example 10.1.15 X = K^2 with rfoo metric: doo(p,q) = max{|t/i -xi\,\y 2 - x 2 \ ...
10.1 Metric Spaces 141 Example 10.1.19 See Figure 10.5. q G intE but p g" intE. Fig. 10.5. Example 10.1.19 Example 10.1.20 Let X ...
142 10 Basic Topology \ i ,--- P i / i i i i. / Fig. 10.7. Example 10.1.26 Example 10.1.26 N is a neighborhood of P but it is no ...
10.1 Metric Spaces 143 CLOSED OPEN Not CLOSED Not OPEN (o) (d) (o) Fig. 10.9. Example 10.1.32 Definition 10.1.33 E is perfect if ...
144 10 Basic Topology 3qx G Br(p) r\E3qi^p^ d(q,p) = n > 0. 3q 2 eBr(p)DE 9 q 2 ^p. Then, q 2 ^qi- Since q 2 ^ p, r 2 = d(q 2 ...
10.1 Metric Spaces 145 Let (X,d) be a metric space, then The union of a finite collection of open sets is open. The intersectio ...
146 10 Basic Topology E is open relative to Y •» Vp e E 3r > 0 3 Br(p) r\Y CE. E is closed relative to Y •£> Y \ E = Y n E ...
10.2 Compact Sets 147 Example 10.2.2 X = Rk with d 2 metric: E = B 1 (0), for n £ N, Gn = 3^.(0) =>£c U??=i Gn. Example 10.2. ...
148 10 Basic Topology => d{p, qi) < d{p, z) + d(z, qi) < d(p, qi). Contradiction! Therefore, W D V = 0. Thus, V = Br(p) ...
10.2 Compact Sets 149 Remark 10.2.14 Here are some remarks: 7/lim„_loo(6n - an) = lim^^ length(In) = 0, => f£°=i In consists ...
150 10 Basic Topology Remark 10.2.19 Consider the following remarks on Theorem 10.2.18: The equivalence of (a) and (b) is known ...
1 < \{(3T - 2 - aT + 1] = ^ ")3" 10.4 Connected Sets 151 1 a3~"3 1 n - 1 a — > ^ > — > 1, is what we want. So, a > ...
152 10 Basic Topology Proof. Let us mark the statement 2, x £ E and x < z => Vy with x<y<z=$-y€E (*). Let E ^ 0 be c ...
10.4 Problems 153 Fig. 10.12. The PCB example machine with a robot arm makes vias (a kind of drill operation) at points A, B,... ...
154 10 Basic Topology robot configuration is to be selected when the opportunity cost of keeping the system busy is C 0 per unit ...
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