Question

Every metric space is normal. (If \(A, B\) are closed sets in the metric space \((X, \rho)\). consider the sets of points \(x\) where \(\rho(x, A)<\rho(x, B)\) or \(\rho(x, A)>\rho(x, B) .)\)

Short answer

Every metric space is normal because disjoint closed sets can be separated by open sets defined by distance functions.

Step-by-step solution

Understanding the Problem

We must show that in a metric space (X, \( \rho \)), any two disjoint closed sets \( A \) and \( B \) can be separated by neighborhoods. Normallity in topology dictates that for any two disjoint closed sets \( A \), \( B \), there exist open sets \( U \) containing \( A \) and \( V \) containing \( B \), such that \( U \cap V = \emptyset \). We will use the distance function to define these open sets.

Define the Distance Function

For a point \( x \) in metric space \( X \), the distance from \( x \) to the set \( A \) is defined as \( \rho(x, A) = \inf\{ \rho(x, a) : a \in A \} \). This represents the smallest distance from \( x \) to any point in \( A \). Similarly, define \( \rho(x, B) \) for set \( B \). As \( A \) and \( B \) are disjoint and closed, their respective infimum values are strictly positive for any point not belonging to the respective set.

Constructing Open Sets

Consider the set \( U = \{ x \in X : \rho(x, A) < \rho(x, B) \} \). Similarly, define \( V = \{ x \in X : \rho(x, B) < \rho(x, A) \} \). These sets \( U \) and \( V \) are designed to separate \( A \) and \( B \), as any point \( x \) closer to \( A \) than \( B \) belongs to \( U \) and vice versa. By their definition, \( U \) and \( V \) are open sets.

Proving Separation

We need to show \( A \subseteq U \) and \( B \subseteq V \). Since every point in \( A \) has \( \rho(a, A) = 0 \) and \( \rho(a, B) > 0 \), every \( a \in A \) is clearly in \( U \). Similarly for \( B \). Additionally, \( U \cap V = \emptyset \): if a point \( x \) belonged to both, it would imply \( \rho(x, A) < \rho(x, B) \) and \( \rho(x, B) < \rho(x, A) \), which is a contradiction. Thus, \( A \) and \( B \) are separated by these open sets.

Conclusion

Since any two disjoint closed sets in a metric space can be separated by non-intersecting open sets, every metric space is indeed normal. This completes the proof.

Key concepts

normal space

In the world of topology, a space is called normal if any two disjoint closed sets can be separated by open sets. This means that if you have two closed sets that don't overlap, you can find open sets around them that don't touch each other either.
Normality is a particularly interesting property because it implies a level of 'separateness' that's important in understanding how spaces behave. Normal spaces allow for intuitive separation of objects, which is crucial in many mathematical analyses.
When dealing with metric spaces, which have a specific way to measure distance (more on that later), they are always normal. This is because, in a metric space, you can use distances to define those open sets perfectly, ensuring they don't intersect.

open sets

The concept of open sets is significant in both metric spaces and more general topological spaces. Simply put, a set is open if, loosely speaking, you can move a bit in any direction around any point of the set without leaving the set. These open sets form an essential building block in the study of topology.
In metric spaces, open sets have a particularly clear definition. You can think of them as collections of points that contain a whole 'neighborhood' around each one. For example, given a point in a set, you can find a tiny bubble of other points around it, all still within the set. When solving problems like our original exercise, you use these open sets to separate other sets, like the disjoint closed sets, into non-overlapping regions.
This property of openness enables mathematicians to analyze and understand spaces and their behaviors through a lens that prioritizes how objects relate to each other spatially.

distance function

Metric spaces rely heavily on the concept of a distance function, typically denoted as \( \rho \). This function allows us to determine the "distance" between any two points in the space, giving us a way to talk about how 'far apart' things are.
The distance function defines how we can cut up or analyze the space, helping us understand its structure and properties. It determines how we define other concepts, like open and closed sets, which are used to show normality in metric spaces.
In our problem, this distance function is used to compare distances from a point to different sets, like \( A \) and \( B \). By constructing sets of points where this distance comparison holds, open sets can be defined. These sets show whether a point is closer to \( A \) or \( B \), and from there, we can use them to effectively separate \( A \) and \( B \) into distinct regions.