Intro to Analysis in-class_Part_21

Intro to Analysis in-class_Part_21 - 3.2 Open sets and...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3.2 Open sets and closed sets 47 3.2 Open sets and closed sets 3.2.1 Open sets QUESTION: Why are open sets handy? ANSWER: They have wiggle room. Definition 3.2.1. lim x n = L iff > , N, n N = | x n- L | < . This is equivalent in R to a more general definition: U (open interval containing L ) , N such that n N = x n U. REASON: L ( a,b ) = | L- a | , | L- b | > 0. Take to be the smaller of the two. Then | x n- L | < = x n ( a,b ). Other direction is similar. Definition 3.2.2. A set U is open iff every point of U lies in an open interval which is contained in U , i.e., if x U = a,b such that x ( a,b ) U. This means open sets are automatically big: they contain uncountably many points; contain EVERY point between the inf and sup of any subinterval. An open set automat- ically has positive length. Interpretation of defn: Roughly, no point of an open set lies on the boundary....
View Full Document

Page1 / 2

Intro to Analysis in-class_Part_21 - 3.2 Open sets and...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online