{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Intro to Analysis in-class_Part_17

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

This preview shows pages 1–2. Sign up to view the full content.

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. Let A = { x . . . x 2 2 } and B = { x . . . x 2 2 } . Then A is the set of lower bounds of B and B is the set of upper bounds of A .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online