Hey all, can I have a quick check on this?
Problem: Construct a totally disconnected compact set such that where denotes the Lebesgue measure.
Solution Sketch: Let be an enumeration of the rationals. To each rational associate an open interval . Take and . Clearly is compact, while
(aside: it's clear I can get this as close to 1 as I like if I choose my intervals differently)
so . is totally disconnected because it contains no rationals, while connected subsets of with more than 1 point contain rationals (intermediate value property of connected subsets of R).