Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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Let F= Ek, where Eo is the equilateral triangle with base [0,1]
and Ek+1 is constructed by dividing each triangle in Ę into nine equal
closed subtriangles and retaining four of these, with the others being
removed, as shown in the diagram below.
(a) Show that F is non-empty and compact.
(b) Use results from Chapter 4 of Falconer to determine the Hausdorff
and box dimensions of F and to show that 0 < H*(F) < ∞, when
s = dimμ F.
(c) State, with careful justification, whether F is
(i) disconnected;
(ii) totally disconnected.
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Transcribed Image Text:Let F= Ek, where Eo is the equilateral triangle with base [0,1] and Ek+1 is constructed by dividing each triangle in Ę into nine equal closed subtriangles and retaining four of these, with the others being removed, as shown in the diagram below. (a) Show that F is non-empty and compact. (b) Use results from Chapter 4 of Falconer to determine the Hausdorff and box dimensions of F and to show that 0 < H*(F) < ∞, when s = dimμ F. (c) State, with careful justification, whether F is (i) disconnected; (ii) totally disconnected.
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