Problem 28 - An Introduction to Abstract Analysis

Question

Give an example of a set \(A\) in \(\mathbb{R}^{2}\) such that \(\bar{A}=\mathbb{R}^{2}\) and int \(A\) is empty.

Step-by-Step Solution

Answer

Question: Find a set 𝐴 in ℝ² with the given properties: the closure of 𝐴 is equal to ℝ² and the interior of 𝐴 is empty. Answer: The set 𝐴 composed of all the rational points in ℝ² satisfies the required properties.

Step 1: Understanding closure and interior

In this context, the closure of a set 𝐴 refers to the union of 𝐴 and its limit points. In ℝ² space, the limit point 𝑥 of 𝐴 is the point such that every open disk centered at 𝑥 contains a point in 𝐴, although 𝑥 may or may not be in 𝐴. The interior of a set 𝐴 is the largest open subset contained in 𝐴.

Step 2: Finding a set 𝐴 in ℝ² such that no open disk is entirely contained within it

In order for the interior of 𝐴 to be empty, there must not be any open disks that are entirely contained in 𝐴. ℝ² can be represented by coordinate points (𝑥,𝑦) where 𝑥 and 𝑦 are real numbers. We can consider a set 𝐴 composed of all the rational points in ℝ², i.e. (𝑥,𝑦) where both 𝑥 and 𝑦 are rational. Since the set of rational numbers is dense in ℝ, there will not be any open disk entirely contained within 𝐴.

Step 3: Verifying that ℝ² is the closure of 𝐴

Let's consider a point 𝑥=(𝑥₁,𝑥₂) in ℝ² with 𝑥₁ and 𝑥₂ being real numbers. Since the set of rational numbers is dense in ℝ, we can find sequences of rational numbers 𝑟₁ₖ and 𝑟₂ₖ such that: \( r_{1k}\to x_1\), \( r_{2k}\to x_2\) Now we can form the sequence 𝑅ₖ=(𝑟₁ₖ, 𝑟₂ₖ) which is a sequence of points in 𝐴. Since 𝑅ₖ converges to 𝑥, 𝑥 is a limit point of 𝐴. Therefore, every point in ℝ² is a limit point of 𝐴.

Step 4: Concluding that the given set satisfies the required properties

We have found a set 𝐴 in ℝ², such that: - int 𝐴 is empty because every open disk contains points that are not in 𝐴. - 𝑎𝜆𝐴 = ℝ² since every point in ℝ² is either in 𝐴 or a limit point of 𝐴. So, the set 𝐴, composed of all the rational points in ℝ², is an example of a set with the required properties.

Key Concepts

Closure of a set

The concept of a closure of a set in mathematics is essential, especially in abstract analysis. When we talk about closure, we mean finding all the points that "touch" a given set \(A\). In formal terms, the closure of a set \(A\) is the smallest closed set that contains \(A\). This includes not only the points in \(A\) itself but also all its limit points.

If you imagine a set as a collection of points scattered around, the closure is the boundary plus all the inside points. Limit points are those that, no matter how close you get, there's always a point of the set nearby.

• The closure of \(A\) is denoted as \(\bar{A}\).
• It is essentially \(A\) along with its limit points.
• Closure covers all the possible convergence points related to the set.

In the problem, the set \(A\) made up of all rational points in \(\mathbb{R}^2\), has \(\bar{A} = \mathbb{R}^2\). This is because no matter where you go in \(\mathbb{R}^2\), there's a rational point close by. Therefore, the entirety of \(\mathbb{R}^2\) encompasses \(A\) including all its limit points.

Interior of a set

The interior of a set \(A\) can be a slightly more tricky concept at first, but it's essentially about understanding the 'heart' of a set. Think of it like this: the interior is the largest open space that you can find inside a set.

An open set in this context is one that contains a neighborhood, or tiny "bubble" of points completely within the set, around each of its points. The interior of \(A\) is written as \(\text{int } A\) and is the union of all such 'bubbles' within \(A\).

• Interior is like a "buffer zone" around points that remain entirely within \(A\).
• If there are no such bubbles, the set has an empty interior.

In our example, rational coordinate points [(x, y) where both are rational] do not create any open bubble in \(\mathbb{R}^2\). This is because between any two rational points, there are always irrational points, preventing a 'bubble' from existing.

Thus, the interior of \(A\), which is composed of these rational points, is empty.

Dense sets

Dense sets are fascinating because they appear to "fill" any space, despite perhaps not containing every point. For a set \(A\) to be dense in \(\mathbb{R}^2\), every point in \(\mathbb{R}^2\) must either be in \(A\) or be a limit point of \(A\).

This means that for every spot you pick in \(\mathbb{R}^2\), you can come infinitely close to it with points from \(A\) without actually needing \(\text{A}\) to fill every spot. It's like trying to completely fill a room with only certain specific blocks. You never entirely cover the floor, but you get infinitely close.

• Dense sets "visit" every point space can hold.
• They often make it possible for many theoretical mathematics applications.

In the example given, the set of rational points in \(\mathbb{R}^2\) is dense because any point with real coordinates can be approached arbitrarily closely by rational points.

So, although the set of all rational points in \(\mathbb{R}^2\) is not the same as \(\mathbb{R}^2\), it is dense in \(\mathbb{R}^2\). This is why its closure is \(\mathbb{R}^2\). As a result, the rational points are scattered so fully across \(\mathbb{R}^2\) that they touch every point.