Because r a this shows p a1 2 this argument holds for

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: hyperlinked to its proof, allowing the general structure of ideas to be seen quickly, with details available whenever desired. Not being able to figure out a physical realization of a hyperlink, we settled for this. First, we establish a result necessary for proving the Hausdorff distance h satisfies the triangle inequality. Lemma 2.2.1 For any compact set A and for any positive numbers δ1 and δ2 (Aδ1 )δ2 ⊆ Aδ1 +δ2 40 CHAPTER 2. ITERATED FUNCTION SYSTEMS With this, we prove the Hausdorff distance is a metric. Proposition 2.2.1 The Hausdorff distance is a metric. Proofs Proof of Lemma 2.2.1. For every point p ∈ (Aδ1 )δ2 , we show p ∈ Aδ1 +δ2 . First, from the definition of thickening we see p ∈ (Aδ1 )δ2 implies there is a point q ∈ Aδ1 with d(p, q ) ≤ δ2 . Next, q ∈ Aδ1 implies there is a point r ∈ A with d(q, r) ≤ δ1 . By the triangle inequality for d, d(p, r) ≤ d(p, q ) + d(q, r) and so d(p, r) ≤ δ1 + δ2 . Because r ∈ A, this shows p ∈ Aδ1 +δ2 . This argu...
View Full Document

This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online