Question

If \(E\) is a nonempty subset of a metric space \(X\), define the distance from \(x \in X\) to \(E\) by $$\rho_{x}(x)=\inf _{x=8} d(x, z) .$$ (a) Prove that \(\rho_{E}(x)=0\) if and only if \(x \in E\). (b) Prove that \(\rho_{E}\) is a uniformly continuous function on \(X\), by showing that $$\left|\rho_{E}(x)-\rho_{E}(y)\right| \leq d(x, y)$$ for all \(x \in X, y \in X\).

Short answer

Based on the given step by step solution, create a short answer as follows: To show that the distance function \(\rho_E\) is uniformly continuous, first we prove that \(\rho_E(x)=0\) implies \(x \in E\) and \(x \in E\) implies \(\rho_E(x)=0\). Then, by applying the triangle inequality, we show that \(\left|\rho_{E}(x)-\rho_{E}(y)\right| \leq d(x, y)\) for all \(x, y \in X\). This proves that \(\rho_E\) is uniformly continuous on \(X\).

Step-by-step solution

Prove that \(\rho_E(x)=0\) implies \(x \in E\):

Assume \(\rho_E(x) = 0\). By definition, \(\rho_E(x) = \inf_{z \in E} d(x, z)\). Since \(\rho_E(x)=0\), it implies that for every \(\varepsilon > 0\), there exists \(z \in E\) such that \(d(x, z) < \varepsilon\). Now, if we take \(\varepsilon\) to be an arbitrarily small positive number, we see that \(d(x, z)\) must be equal to zero for some \(z \in E\). Thus, \(x=z\) and \(x \in E\).

Prove that \(x \in E\) implies \(\rho_E(x)=0\):

Assume \(x \in E\). Since \(d(x, x) = 0\) (by the properties of metric spaces), it follows that \(\rho_E(x) \leq d(x, x) = 0\), and since the distance cannot be negative, we have \(\rho_E(x) = 0\).

Prove that \(\left|\rho_{E}(x)-\rho_{E}(y)\right| \leq d(x, y)\):

Let \(x, y \in X\). For any \(z \in E\), we apply the triangle inequality to the distances \(d(x, z)\), \(d(x, y)\), and \(d(y, z)\): $$d(x, z) \leq d(x, y) + d(y, z)$$ Therefore, \(d(x, z) - d(x, y) \leq d(y, z)\), and by the definition of \(\rho_E(y)\), we have: $$\rho_E(y) \leq d(x, z) - d(x, y)$$ Now, taking the infimum over all \(z\in E\), we get: $$\rho_E(y) \leq \rho_E(x) - d(x, y)$$ Similarly, we can show that $$\rho_E(x) \leq \rho_E(y) - d(y, x)$$ Since \(d(x, y) = d(y, x)\), both inequalities can be combined to get: $$\left|\rho_E(x) - \rho_E(y)\right| \leq d(x, y)$$ This shows that \(\rho_E\) is uniformly continuous on \(X\), proving part (b) of the exercise.

Key concepts

Metric Space

A metric space is a mathematical structure that allows us to define a concept of distance between any two elements in a set. The components of a metric space consist of a set \(X\) and a metric (or distance function) \(d\). The metric \(d\) evaluates the distance between any two points in the set and must satisfy certain properties:

• Non-negativity: For any points \(x\) and \(y\) in \(X\), the distance \(d(x, y)\) must be a non-negative value, zero or greater: \(d(x, y) \geq 0\).


• Identity of Indiscernibles: The distance \(d(x, y)\) is zero if and only if the two points \(x\) and \(y\) are identical: \(d(x, y) = 0\) iff \(x = y\).


• Symmetry: The distance between two points is the same irrespective of the order: \(d(x, y) = d(y, x)\).


• Triangle Inequality: For any points \(x\), \(y\), and \(z\) in \(X\), the direct distance from \(x\) to \(z\) should not exceed the sum of the distances from \(x\) to \(y\) and from \(y\) to \(z\): \(d(x, z) \leq d(x, y) + d(y, z)\).

In essence, a metric space gives us a formal way to understand how elements in a set relate to one another in terms of "distance."

Distance Function

A distance function, or metric, is a critical component within a metric space that assigns numerical values representing the distance between any two points within a set. We denote this function as \(d\), with \(d(x, y)\) expressing the distance between point \(x\) and point \(y\).

• A valid distance function adheres to the four properties mentioned in the metric space discussion: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.


• This function essentially quantifies the "gap" between points, offering a measure to determine how far apart two elements are in the set.


• It is important to note that, for any given metric space, the choice of distance function can influence how continuity and convergence are understood within that space.

The role of the distance function is pivotal for analyzing various concepts in analysis and topology, such as continuity, limits, and compactness.

Triangle Inequality

The triangle inequality is a fundamental property of the distance function in a metric space, reflecting the intuitive notion of distance adherence. Specifically, it suggests that for any three points \(x\), \(y\), and \(z\) within a metric space, the shortest path from \(x\) to \(z\) is not greater than the sum of the paths from \(x\) to \(y\) and \(y\) to \(z\). Mathematically, this is represented as:
\[d(x, z) \leq d(x, y) + d(y, z)\]
Understanding the triangle inequality enables us to deduce many essential properties within metric spaces:

• Verification of Uniform Continuity: One useful application of the triangle inequality is demonstrating the uniform continuity of functions. For example, showing \(\left|\rho_E(x) - \rho_E(y)\right| \leq d(x, y)\) as per our exercise verifies uniform continuity.


• Path Development: It helps affirm that direct distances remain the shortest, guiding concepts in pathfinding and optimization problems.


• Consistency Maintenance: Ensures that distance measurements through intermediary points remain logical and consistent within the space.

The triangle inequality is vital in various mathematical theories and applications, as it effectively governs the integrity of distance calculations within metric spaces, ensuring that relationships among points are consistently defined.