The Cantor set is a classic example of a set that is uncountably infinite, has zero measure, and exhibits some counterintuitive properties in terms of size and density. It is constructed through an iterative process starting with the closed interval \([0, 1]\). Here’s how the construction works: 1. **Start with the interval**: Begin with the closed interval \([0, 1]\).
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The Cantor set is obtained from by repeatedly deleting the open middle third of every remaining interval. Equivalently, it consists of the numbers that admit a base-three expansion using only the digits zero and two.