Intro to Analysis in-class_Part_17

Intro to Analysis in-class_Part_17 - Chapter 3 Topology of...

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Chapter 3 Topology of the Real Line “Topology”: the study of qualitative geometric properties: connected, continuous, . .. 1 Abstractly, this amounts to studying open sets. In R , this is unions of intervals ( a,b ). In analysis, topology is all about knowing when a sequence converges. x n L ⇐⇒ ( U is an open nbd of L = x n U for all but finitely many n ) . 3.1 Limits and bounds Not all sequences have a limit, so we need another idea. Definition 3.1.1. Let A be a nonempty subset of R . Then define the supremum of A to be the least (smallest) upper bound of A . In other words, a = sup A means: 1. A a , that is, x A = x a . So a is an upper bound of A . 2. A b = a b . So a is the smallest upper bound. If A has no upper bound, write sup A = . Definition 3.1.2. The infimum of a nonempty set A R is the greatest lower bound of A , defined analogously. Example 3.1.1.
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Intro to Analysis in-class_Part_17 - Chapter 3 Topology of...

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