Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Take \(\mathbb{Q}\) with the standard metric, \(d(x, y)=|x-y|,\) as our metric space. Prove that \(\mathbb{Q}\) is totally disconnected, that is, show that for every \(x, y \in \mathbb{Q}\) with \(x \neq y\), there exists an two open sets \(U\) and \(V\), such that \(x \in U, y \in V, U \cap V=\emptyset,\) and \(U \cap V=\mathbb{Q}\).

Step-by-Step Solution

Answer

\(\mathbb{Q}\) is totally disconnected because for any \(x \neq y \in \mathbb{Q}\), we can find disjoint open sets separating them.

Step 1: Understand the Definition of Totally Disconnected

A space is totally disconnected if the only connected subsets are singletons. In other words, for any two distinct points, there exists a separation by disjoint open sets.

Step 2: Choose Points in the Space

Consider two distinct rational numbers, say \(x\) and \(y\) with \(x eq y\). Without loss of generality, assume \(x < y\).

Step 3: Find the Midpoint

Calculate the midpoint \(m\) of \(x\) and \(y\) using the formula \(m = \frac{x + y}{2}\). This point \(m\) is a rational number since it is the average of two rational numbers.

Step 4: Define Open Sets U and V

Define open sets \(U = \{ z \in \mathbb{Q} : z < m \}\) and \(V = \{ z \in \mathbb{Q} : z > m \}\). These sets are open in the standard metric on \(\mathbb{Q}\), because for any \(z \in U\) or \(z \in V\), there exists a small \(\epsilon > 0\) such that the neighborhood \((z - \epsilon, z + \epsilon)\) is contained within \(U\) or \(V\) respectively.

Step 5: Verify the Properties of U and V

Check that \(x \in U\) and \(y \in V\). Since \(x < m\) and \(y > m\), this condition is satisfied. Also, \(U \cap V = \emptyset\) because no rational number can be simultaneously less than and greater than \(m\).

Step 6: Conclude \(\mathbb{Q}\) is Totally Disconnected

Since for any distinct \(x\) and \(y\) there exists such disjoint open sets \(U\) and \(V\) with \(x \in U\) and \(y \in V\), the rational numbers \(\mathbb{Q}\) are totally disconnected in the metric space sense.

Key Concepts

Rational Numbers

Rational numbers, commonly denoted as \(\mathbb{Q}\), are numbers that can be expressed as the ratio of two integers, where the numerator and denominator are integers and the denominator is not zero. Often, they are in the form \(\frac{a}{b}\) where \(a\) and \(b\) belong to \(\mathbb{Z}\), and \(b eq 0\).

Examples of rational numbers include \(1/2\), \(3\), and \(-5/4\). Note that an integer itself, such as \(3\), is a rational number since it can be written as \(3/1\).

In terms of their position on the number line, rational numbers are dense, meaning between any two rational numbers, there is always another rational number. This property is significant because it suggests that the rational numbers are more finely spaced compared to integers, yet they are still a countable set, unlike the real numbers which are uncountable.

The idea of rational numbers being totally disconnected, as seen in the context of this exercise, relates to their separation via certain properties in metric spaces, especially in the standard metric which we'll discuss next.

Standard Metric

The standard metric is a way to measure distance between two points in a space, and for rational numbers, it is represented by the absolute difference. For any two rational numbers \(x\) and \(y\), the distance between them on the number line can be calculated using the formula \(d(x, y) = |x - y|\).

Here, the vertical bars \(|x - y|\) signify the absolute value, which returns the non-negative difference between \(x\) and \(y\).

This distance function is a crucial component in defining the metric space structure for rational numbers \(\mathbb{Q}\).

• It allows us to consider when two rational numbers are close or far apart.
• It lays the groundwork for designing open sets, such as those used in assessing disconnection of a space.

The simplicity of this metric makes it easy to work with, providing an intuitive understanding of distance in rational spaces. It also plays a key role in defining open sets which help in visualizing the truly disconnected nature of \(\mathbb{Q}\).

Open Sets

Open sets are fundamental in topology and in defining a totally disconnected space. In the context of \(\mathbb{Q}\) with the standard metric, an open set is essentially a collection of points that includes all points within a certain distance from any point inside it.

For example, consider an open interval \((a, b)\) on the number line. This interval consists of all points \(x\) where \(a < x < b\). In this case, \((a, b)\) doesn't include the points \(a\) and \(b\) themselves, which makes it "open."

• In the exercise, two key open sets are \(U = \{ z \in \mathbb{Q} : z < m \}\) and \(V = \{ z \in \mathbb{Q} : z > m \}\).
• Each of these sets contains rational numbers separated by the midpoint \(m\).

Understanding open sets helps us visually separate rational numbers into distinct regions, highlighting how spaces can be constructed to show disconnection between points.

Midpoint Calculation

Midpoint calculation is an essential concept when discussing separating points in a metric space composed of rational numbers. Finding the midpoint helps in dividing the space into two disjoint open sets.

Given any two rational numbers \(x\) and \(y\), the midpoint is calculated using\(m = \frac{x + y}{2}\). This formula ensures that \(m\) is a number right in between \(x\) and \(y\).

Since \(x\) and \(y\) are rational, \(m\) is also rational. This is due to the closure property of rational numbers, which states that the sum and averaged value of rational numbers is always a rational number.

• Calculating the midpoint enables the creation of new open sets like \(U = \{ z \in \mathbb{Q} : z < m \}\) and \(V = \{ z \in \mathbb{Q} : z > m \}\).
• These sets are critical for proving the concept of total disconnection among rational numbers.

By dividing the space with the midpoint, we ensure there's a clean separation, facilitating a key understanding of connectedness and disconnectedness in metric spaces.