Question

Metrize the extended real number system \(E^{*}\) by $$ \rho^{\prime}(x, y)=|f(x)-f(y)|, $$ where the function $$ f: E^{*} \underset{\text { onto }}{\longrightarrow}[-1,1] $$ is defined by \(f(x)=\frac{x}{1+|x|}\) if \(x\) is finite, \(f(-\infty)=-1,\) and \(f(+\infty)=1\) Compute \(\rho^{\prime}(0,+\infty), \rho^{\prime}(0,-\infty), \rho^{\prime}(-\infty,+\infty), \rho^{\prime}(0,1), \rho^{\prime}(1,2),\) and \(\rho^{\prime}(n,+\infty)\). Describe \(G_{0}(1), G_{+\infty}(1),\) and \(G_{-\infty}\left(\frac{1}{2}\right)\). Verify the metric axioms (also when infinities are involved).

Short answer

The distances are: 1, 1, 2, 0.5, 1/6, 1/(1+n). The sets are: all finite numbers, same excluding -∞, empty.

Step-by-step solution

Compute \( \rho^{\prime}(0, + fty) \)

To find \( \rho^{\prime}(0, +\infty) \), we use the function \( f \) given by \( f(0) = \frac{0}{1+|0|} = 0 \) and \( f(+\infty) = 1 \). Thus, \( \rho^{\prime}(0, +\infty) = |f(0) - f(+\infty)| = |0 - 1| = 1 \).

Compute \( \rho^{\prime}(0, -\infty) \)

Here, \( f(0) = 0 \) as before, and \( f(-\infty) = -1 \). Calculate \( \rho^{\prime}(0, -\infty) = |f(0) - f(-\infty)| = |0 + 1| = 1 \).

Compute \( \rho^{\prime}(-\infty, +\infty) \)

Compute using \( f(-\infty) = -1 \) and \( f(+\infty) = 1 \). So, \( \rho^{\prime}(-\infty, +\infty) = |f(-\infty) - f(+\infty)| = |-1 - 1| = 2 \).

Compute \( \rho^{\prime}(0, 1) \)

Using \( f(0) = 0 \) and \( f(1) = \frac{1}{1+1} = \frac{1}{2} \), we find \( \rho^{\prime}(0, 1) = |0 - \frac{1}{2}| = \frac{1}{2} \).

Compute \( \rho^{\prime}(1, 2) \)

Compute \( f(1) = \frac{1}{2} \) and \( f(2) = \frac{2}{1+2} = \frac{2}{3} \). Then, \( \rho^{\prime}(1, 2) = |\frac{1}{2} - \frac{2}{3}| = \frac{1}{6} \).

Compute \( \rho^{\prime}(n, +\infty) \)

For any finite \( n \), \( f(n) = \frac{n}{1+n} \) and \( f(+\infty) = 1 \). Then, \( \rho^{\prime}(n, +\infty) = \left|\frac{n}{1+n} - 1\right| = \frac{1}{1+n} \).

Describe \( G_{0}(1) \)

\( G_{0}(1) \) is the set of all points \( y \) such that \( \rho^{\prime}(0, y) < 1 \). This includes all finite \( y \) and excludes \(+\infty\) and \(-\infty\), since \( \rho^{\prime}(0, \pm\infty) = 1 \).

Describe \( G_{+\infty}(1) \)

\( G_{+\infty}(1) \) consists of all points \( y \) such that \( \rho^{\prime}(y, +\infty) < 1 \). These points includes all finite numbers and \(+\infty\), but not \(-\infty\) because \( \rho^{\prime}(y, -\infty) \geq 1 \).

Describe \( G_{-\infty}\left( \frac{1}{2} \right) \)

In this case, \( G_{-\infty}\left( \frac{1}{2} \right) \) is the set of all \( y \) such that \( \rho^{\prime}(y, -\infty) < \frac{1}{2} \). No finite \( y \) satisfies this since \( \rho^{\prime}(y, -\infty) = 1 \) or greater. Thus, it includes no points.

Verify Metric Axioms

1. Non-negativity: \( \rho^{\prime}(x, y) \geq 0 \) for all \( x, y \).2. Identity of indiscernibles: \( \rho^{\prime}(x, y) = 0 \) if and only if \( x = y \).3. Symmetry: \( \rho^{\prime}(x, y) = \rho^{\prime}(y, x) \).4. Triangle inequality: \( \rho^{\prime}(x, z) \leq \rho^{\prime}(x, y) + \rho^{\prime}(y, z) \). All conditions hold, including when infinities are involved.

Key concepts

Extended Real Number System

The extended real number system, denoted by \( E^{*} \), is essentially an augmentation of the traditional real number set \( \mathbb{R} \) by including two additional elements: \(+\infty\) and \(-\infty\). This expanded set is useful in various branches of mathematics, particularly when dealing with limits and integrals of unbounded functions.
The inclusion of \(+\infty\) and \(-\infty\) helps in providing a comprehensive framework for describing limits. This allows for more inclusive analysis, as all infinite behaviors are encapsulated within the number system.
- \(+\infty\): Representing the concept of positive infinity, it is larger than all real numbers.- \(-\infty\): Signifying negative infinity, it is lesser than all real numbers.
In problems involving the distance between points in \( E^{*} \), infinity is treated as a distinct point. For example, using the function \( f \) provided, we can compute distances involving these infinities to better understand how they behave in the context of metric spaces.

Metric Axioms

Metric spaces are mathematical structures used to define the notion of distance between points. Any function defined to measure this distance, \( \rho^{\prime}(x, y) \), must satisfy specific properties known as metric axioms.

• Non-negativity: \( \rho^{\prime}(x, y) \geq 0 \). The distance is always non-negative.
• Identity of Indiscernibles: \( \rho^{\prime}(x, y) = 0 \) if and only if \( x = y \). Only identical points have zero distance between them.
• Symmetry: \( \rho^{\prime}(x, y) = \rho^{\prime}(y, x) \). The distance from \( x \) to \( y \) is the same as from \( y \) to \( x \).
• Triangle Inequality: \( \rho^{\prime}(x, z) \leq \rho^{\prime}(x, y) + \rho^{\prime}(y, z) \). The direct path between two points is never longer than any indirect path.

These axioms ensure that the distance function behaves in a consistent and predictable way, accommodating both finite numbers and infinities.

Distance Function

In the context of metric spaces, a distance function \( \rho \) is crucial for quantifying how far apart two points are. In this exercise, the distance function \( \rho^{\prime}(x, y) = |f(x) − f(y)| \) is used, with \( f \) mapping elements from \( E^{*} \) to the interval \([-1,1]\). This setup is particularly useful in measuring distances in systems involving infinities.
- The function \( f \) is defined as \( f(x) = \frac{x}{1 + |x|} \) for finite \( x \), \( f(-\infty) = -1 \), and \( f(+\infty) = 1 \). This mapping compresses the infinite range into a more manageable bounded form, allowing us to apply standard metric axioms.
When evaluating \( \rho^{\prime} \) values such as \( \rho^{\prime}(0, +\infty) \) or \( \rho^{\prime}(-\infty, +\infty) \), the calculation involves straightforward substitution into \( |f(x) − f(y)| \), thus simplifying operations involving potentially challenging infinite elements.

Set Theory

Set theory is an essential part of understanding metric spaces as it involves the study and analysis of collections of objects, here being numbers in our metric space. In this context, we examine specific sets such as \( G_{0}(1) \), \( G_{+\infty}(1) \), and \( G_{-\infty}(\frac{1}{2}) \).
- \( G_{0}(1) \) is defined as all points \( y \) for which \( \rho^{\prime}(0, y) < 1 \), essentially capturing all finite \( y \), excluding \( +\infty \) and \( -\infty \).
- \( G_{+\infty}(1) \) includes points satisfying \( \rho^{\prime}(y, +\infty) < 1 \). This covers all finite points and \( +\infty \), omitting \( -\infty \) as the distance is inherently \( 2 \) or more from \( +\infty \).
- \( G_{-\infty}(\frac{1}{2}) \) needs \( \rho^{\prime}(y, -\infty) < \frac{1}{2} \), though no finite point meets this criterion because the minimum distance is \( 1 \).
These sets help illustrate different regions of the extended real number system, adding depth to our understanding of how points relate within a metric context.