Question

Let \(T\) be a compact subset of a metric space \((A, \rho) .\) Show that there are members \(\bar{S}\) and \(\bar{t}\) of \(T\) such that \(d(\bar{s}, \bar{t})=d(T)\).

Short answer

Question: Show that there exist points \(\bar{s}\) and \(\bar{t}\) in a compact subset \(T\) of a metric space \((A, \rho)\) such that the distance between them is the distance between the subset \(T\) itself, i.e., \(d(\bar{s}, \bar{t}) = d(T)\). Answer: We defined the maximum distance between two points in \(T\) as \(D = \sup_{s,t\in T} \rho(s,t)\), and let \(f(s)=d(s,T)\) be the distance between a point \(s\) and the subset \(T\). Since \(T\) is compact and \(f\) is continuous, we found the maximum value of \(f\) on \(T\) at the point \(\bar{s}\). Then, we chose a point \(\bar{t}\) in \(T\) such that \(D/2 \leq d(\bar{s}, \bar{t})\). By deriving the inequalities related to the metric properties and the compactness of \(T\), we concluded that \(d(\bar{s}, \bar{t})=d(T)\), showing the existence of such points \(\bar{s}\) and \(\bar{t}\) in \(T\).

Step-by-step solution

Recall Compactness

A subset \(T\) of a metric space \((A, \rho)\) is called compact if every open cover of \(T\) has a finite subcover, i.e., for any collection of open sets \(\{U_i\}\) that covers \(T\), there exists a finite subcollection \(\{U_1, U_2, ..., U_n\}\) that also covers \(T\).

Recall Distance in Metric Space

The distance between two points \(s\) and \(t\) in a metric space \((A, \rho)\) is given by the function \(\rho(s, t)\). The distance between a point \(s \in A\) and a subset \(T \subset A\) is given by \(d(s, T) := \inf_{t \in T}\rho(s, t)\). The distance between two subsets \(S, T \subset A\) is defined as \(d(S, T) := \inf_{s\in S, t\in T} \rho(s,t)\).

Find the maximum distance in T

First, let's find the maximum distance between two points in T. Define \(D := \sup_{s,t\in T} \rho(s,t)\). Since \(T\) is compact, \(D\) is finite.

Construct the function f(s)

For each \(s \in T\), let \(f(s) := d(s, T)\). The function \(f: T \to \mathbb{R}_{\geq 0}\) is continuous since it involves taking infimums of continuous functions (\(\rho\)).

Use Compactness to find the maximum of f(s)

As \(T\) is compact and \(f\) is continuous, the image of \(f(T)\) is also compact in \(\mathbb{R}\). Hence, \(f\) achieves a maximum on \(T\). Let \(\bar{s} \in T\) be the point at which \(f\) achieves its maximum, i.e., \(f(\bar{s}) = \sup_{s \in T} f(s)\).

Construct the point \(\bar{t}\)

Since \(\bar{s}\) is the point at which \(f\) achieves its maximum, we have \(f(\bar{s}) \geq f(t)\) for all \(t \in T\). So, \(d(\bar{s}, T) = f(\bar{s}) = \inf_{t \in T}\rho(\bar{s}, t) \geq d(\bar{s}, \bar{t})\). Now, let's choose a point \(\bar{t} \in T\) such that \(d(\bar{s}, \bar{t}) \geq D/2\).

Derive Contradiction and Conclude

Now we have \(d(\bar{s}, T) \geq d(\bar{s}, \bar{t}) \geq D/2\). Taking infimum over \(s \in T\), we obtain \(d(T) \geq d(\bar{s}, \bar{t}) \geq D/2\), which implies that \(D \leq 2d(T)\). On the other hand, by the definition of \(D\), we have \(d(T) = d(\bar{s}, \bar{t}) \leq D\). Therefore, \(2d(T) \leq D\). Combining these two inequalities, we get \(D = 2d(T)\). Since \(d(\bar{s}, \bar{t}) \geq d(T)\) and \(D = 2d(T) \geq d(\bar{s}, \bar{t})\), we have \(d(\bar{s}, \bar{t}) = d(T)\). Thus, we have found the points \(\bar{s}\) and \(\bar{t}\) in \(T\) such that \(d(\bar{s}, \bar{t})=d(T)\) as desired.

Key concepts

Metric Space Distance

In mathematics, understanding metric spaces plays a crucial role in analysis. A metric space is a set equipped with a function that measures the 'distance' between any two elements within that set. This distance function, usually denoted as \( \rho \), must satisfy four specific properties:

• Non-negativity: \( \rho(s, t) \geq 0 \) for all points \( s, t \).
• Identity of indiscernibles: \( \rho(s, t) = 0 \) if and only if \( s = t \).
• Symmetry: \( \rho(s, t) = \rho(t, s) \).
• Triangle inequality: \( \rho(s, t) \leq \rho(s, u) + \rho(u, t) \) for any points \( s, t, u \).

The key aspect here is how we can measure and derive meaningful properties of subsets within these spaces. Specifically, the distance between a point \( s \) and a subset \( T \) is defined as the smallest (infimum) possible distance from \( s \) to any point in \( T \). When we're considering two subsets \( S \) and \( T \), the distance is the smallest possible distance between any pair of points with one in \( S \) and the other in \( T \). This understanding is vital for studying more complex properties of metric spaces, such as compactness.

Compact Subsets

A central topic in analysis is compactness, a concept which extends the idea of closed and bounded sets from Euclidean spaces to more abstract spaces like metric spaces. A subset \( T \) of a metric space \( (A, \rho) \) is compact if every open cover of \( T \) has a finite subcover. Simply put, this means if you can cover \( T \) with a collection of open sets, you can also cover it with just a finite number of those sets, which is particularly useful because it is like saying the space, in some sense, is 'limited' or 'contained'.
Compactness is a fundamental property because it ensures certain key features of functions and spaces:

• Continuity: Any continuous function defined on a compact space is uniformly continuous.
• Boundedness: Any compact set in a metric space is also bounded.
• Closed: Compact subsets of metric spaces are closed.
• Finite subcovering: As noted, the definition itself hinges on being able to cover the set with a finite union of open sets.

These properties are essential in various branches of mathematics, such as when proving the existence of maximum or minimum values for real-valued continuous functions defined on compact sets.

Continuous Functions in Real Analysis

Continuous functions are the key players in real analysis and metric spaces. A function \( f \) is deemed continuous if small changes (or disturbances) to the input result in small changes in the output. Formally, a function \( f : A \rightarrow B \) is continuous at a point \( c \in A \) if, for every \( \epsilon > 0 \), there is a \( \delta > 0 \) such that whenever \( \rho_A(x, c) < \delta \) (where \( x \) is a point in \( A \)), it follows that \( \rho_B(f(x), f(c)) < \epsilon \). This concept effectively translates everyday notions of unbroken, steady transformation from one set to another.
In the context of metric spaces and compact subsets, continuous functions exhibit noteworthy properties:

• Image of a Compact Set: The image of a compact set under a continuous function is itself compact. This implies that if you start with a space having the compactness property, this will be preserved when transformed by a continuous function.
• Maximum and Minimum Values: For continuous real-valued functions defined on compact spaces, the function must attain a maximum and minimum. This principle is instrumental in optimization problems, where finding extrema is necessary.
• Connection with Uniformity: On a compact space, a continuous function is uniformly continuous, meaning the typical \( \epsilon-\delta \) condition holds uniformly over the entire domain.

Understanding these characteristics is fundamental for delving deeper into real analysis and solving complex problems involving function behavior and analysis in metric spaces.