Question

Prove the assertion that the Cantor set is nowhere dense

Short answer

The Cantor set is nowhere dense, because every interval \(E\) contained in it can always be 'stretched' to contain a point not in the Cantor set \(C\), hence \(C\) does not contain any open intervals and is nowhere dense in [0,1].

Step-by-step solution

Identify Cantor set

A Cantor set, in this problem, is conceived through a process of infinite iteration. Initialize with the closed interval [0,1]. For all steps after the first, take the remaining collection of closed intervals and remove the open middle thirds. This process will generate a Cantor set.

Understanding 'nowhere dense' set

A set in a topological space is called 'nowhere dense' if the closure of the set has empty interior. In simpler terms, the set and its boundary does not contain any open intervals.

Prove Cantor set is 'nowhere dense'

The Cantor set \(C\) consists of all points in the interval [0,1] that have a ternary expansion (an expansion in base 3) consisting only of 0s and 2s. Consider an interval \(E\) contained in \(C\). This implies that every ternary decimal in \(E\) must consist exclusively of 0s and 2s. However, one could always construct a ternary decimal in \(E\) that contains a 1 by choosing a digit in the decimal expansion to be a 1, meaning this decimal is not a member of \(C\). This shows that any interval \(E\) can contain elements not in \(C\), implying the Cantor set is 'nowhere dense'.

Key concepts

Nowhere Dense

When we talk about a set being "nowhere dense" in a topological space, we refer to a particular feature concerning its size and spread. A set is considered nowhere dense if, within the space it's situated, you cannot find any interval that is open, fully inside the closure of the set.
This concept might sound a bit tricky at first, but here's a way to picture it:

• Imagine planting trees in a forest. If you plant so sparsely that no path between locations allows travelers to move entirely within areas covered by trees, then your planting is nowhere dense.
• There's no continuous stretch where folks can walk that is fully covered by your trees.

For the Cantor set, being nowhere dense means it cannot "fill" any part of the interval [0,1] with open intervals. You can't take a stroll in the Cantor set without stepping off it again.

Topological Space

Before we move on with understanding sets like the Cantor set, it's vital to grasp what a topological space is. A topological space is essentially a set that, in addition to its elements, has a collection of open sets that define its structure. These open sets help mathematicians discuss concepts like limits, closure, and continuity.

• Picture your house and other houses on the neighborhood map: the open sets here are like groupings of houses connected by streets without any boundaries.
• A topological space isn't about the houses (or points) themselves but rather how they're arranged.

Here, the interval [0,1] can be part of a topological space where each element's relationship with others plays a crucial role in defining properties like 'nowhere dense.'

Ternary Expansion

Ternary expansion is merely another way to express numbers, specifically using base 3 instead of the more common base 10 (decimal system). In the context of the Cantor set, numbers are represented using only the digits 0 and 2. This means that no place is taken by the digit 1.

• Imagine a language where every word is built without using the letter "A."
• For the Cantor set, removing "A" is like removing "1" from numbers that could be between 0 and 1.

Thus, each number in the Cantor set has a unique ternary form without the digit "1," contributing to its fragmented or discrete nature within the interval.

Mathematical Proof

Mathematical proof is the process of showing that certain statements are universally true based on accepted definitions and logical deductive reasoning. In our exercise, proving the Cantor set is nowhere dense involves several steps.

• Firstly, define the Cantor set clearly, understanding its construction through removing open middle thirds from intervals iteratively.
• Next, interpret the concept of nowhere dense by examining how no open interval can be fully contained within the Cantor set.

Using these logical steps, we demonstrate that even though the Cantor set fills everywhere within [0,1], it never forms a continuous block or an open interval entirely inside it.
This stepwise logical argument helps in cementing the property's truth universally across all scenarios involving the Cantor set.