Question

(a) If \(A\) and \(B\) are disjoint closed sets in some metric space \(X\), prove that they are separated. (b) Prove the same for disjoint open sets. (c) Fix \(p \in X, 8>0\), define \(A\) to be the set of all \(q \in X\) for which \(d(p, q)<\delta\), define \(B\) similarly, with \(>\) in place of \(<\). Prove that \(A\) and \(B\) are separated. (d) Prove that every connected metric space with at least two points is uncountable. Hint: Use (c).

Short answer

Question: Prove that every connected metric space with at least two points is uncountable, using the fact that there exist separated sets with respective distances less than and greater than a given delta. Answer: Let X be a connected metric space with at least two distinct points p and q. From the properties of separated sets, we know that there exist separated sets A and B defined by their distances from a fixed point p and a given delta. If X is countable, it would be possible to map X to a countable linearly ordered set L through a bijective continuous function. However, due to the separation of A and B, any continuous function from X to another topological space cannot cross from A to B without discontinuity. This means that the function would map elements in A to elements smaller than a certain value in L and elements in B to elements greater than that value. This creates a contradiction, as the function would not be continuous. Therefore, there cannot exist a bijective continuous function from X to a countable linearly ordered set, and X must be uncountable.

Step-by-step solution

Part (a): Disjoint closed sets

If A and B are disjoint closed sets in a metric space X, it means that both A and B are closed subsets of X and their intersection is empty (i.e., \( A \cap B = \emptyset\)). To show that A and B are separated, we need to prove that: 1. \( \overline{A} \cap B = \emptyset \) 2. \( A \cap \overline{B} = \emptyset \) Since A and B are closed sets, it means that their closures are equal to the sets themselves (i.e., \(\overline{A} = A\) and \(\overline{B} = B\)). Therefore, we have: 1. \( A \cap B = \emptyset \) 2. \( A \cap B = \emptyset \) which are true because A and B are disjoint. Hence, A and B are separated.

Part (b): Disjoint open sets

If A and B are disjoint open sets in a metric space X, it means that both A and B are open subsets of X and their intersection is empty (i.e., \( A \cap B = \emptyset\)). We will prove that A and B are separated: 1. Show that \( \overline{A} \cap B = \emptyset \) 2. Show that \( A \cap \overline{B} = \emptyset \) Suppose \( \overline{A} \cap B \neq \emptyset\), then there exists an element x in \( \overline{A} \cap B\). Since x is in B and B is open, there exists \( r > 0\) such that the open ball \(B(x, r)\) is a subset of B. But since x is also in \(\overline{A}\), there exists an element y in A such that \(d(x, y) < r\). This implies that y is in \(B(x, r)\) and thus in B, which contradicts the fact that A and B are disjoint. Therefore, \(\overline{A} \cap B = \emptyset\). Similarly, we can show that \( A \cap \overline{B} = \emptyset\), and hence A and B are separated.

Part (c): Separated sets with distance conditions

We define two sets A and B as follows: - A = {q ∈ X | \( d(p, q) < \delta \)} - B = {q ∈ X | \( d(p, q) > \delta \)} We need to prove that A and B are separated. Since A and B do not intersect by definition, we can use the results from parts (a) and (b) to show that their closures also do not intersect. 1. Show that \( \overline{A} \cap B = \emptyset \) 2. Show that \( A \cap \overline{B} = \emptyset \) First, suppose there exists an element x in \( \overline{A} \cap B\). This means that x is in B, so \(d(p, x) > \delta\), and x is in \(\overline{A}\), so there exists an element y in A such that \(d(x, y) < \epsilon\) for any \(\epsilon > 0\). Then, using the triangle inequality: \( d(p, y) \le d(p, x) + d(x, y) < \delta + \epsilon\) Since \(\epsilon\) can be arbitrarily small, we can choose it such that \( \delta + \epsilon = d(p, x)\), which contradicts that \(d(p, y) < \delta\). Therefore, \( \overline{A} \cap B = \emptyset\). Similarly, we can show that \( A \cap \overline{B} = \emptyset\), and hence A and B are separated.

Part (d): Connected metric spaces are uncountable

Let X be a connected metric space with at least two distinct points p and q. We want to show that X is uncountable. From part (c), we know that there exist separated sets A and B with respective distances less than and greater than \(\delta\) for any fixed \(p \in X\) and \(\delta > 0\). Let's define \( \delta = \frac{1}{2} d(p, q)\). Then, we construct the following sets: - A = {x ∈ X | \( d(p, x) < \delta \)} - B = {x ∈ X | \( d(p, x) > \delta \)} Since A and B are separated, any continuous function from X to another topological space has the property that the function cannot cross from A to B without discontinuity. If X is countable, then there exists a bijective continuous function from X to a countable linearly ordered set L. However, this would mean that the function would map elements in A to elements smaller than a certain value in L, and elements in B to elements greater than that value, which is not possible due to the separation of A and B. Hence, there cannot exist a bijective continuous function from X to a countable linearly ordered set, and thus X must be uncountable.

Key concepts

Metric Space

A metric space is a fundamental concept in mathematical analysis, which forms the foundation for studying the geometry of sets in space. It's essentially a set \(X\) paired with a function \(d: X \times X \rightarrow \mathbb{R}\) – called the metric or distance function – that defines the distance between elements within that set.

The metric must satisfy the following conditions for all \(x, y, z \in X\):

• The distance \(d(x, y) \geq 0\), and \(d(x, y) = 0\) if and only if \(x = y\).
• It is symmetric, which means \(d(x, y) = d(y, x)\).
• The triangle inequality holds, implying \(d(x, z) \leq d(x, y) + d(y, z)\).

For the textbook exercise, understanding metric spaces is crucial because it helps to define separateness and connectedness of sets—which relates directly to the problems posed in parts (a) through (d).

Connected Spaces

Connected spaces are a topological concept stating, in simple terms, that a space is connected if it cannot be divided into two disjoint nonempty open sets. This notion is pivotal in the part (d) of the exercise, which touches upon the fact that every connected metric space with at least two points is uncountable.

For a space to be considered connected, there can be no 'gap' or break between any two points in the space. In the context of the set \(X\), if A and B are separated, there’s no way to transition smoothly from points in A to points in B without encountering a form of discontinuity.

It helps to visualize connectedness as a stretchable, continuous piece of material without tears. If such tears existed, defining distinct sections of the material, it would no longer be connected. This property of connected spaces leads to a profound conclusion that they are, in fact, uncountable, which underpins the logic behind the proof in the textbook exercise.

Separated Sets

Separated sets are those in which two subsets of a metric space do not 'interact' with each other’s closure. In non-technical language, imagine two circles in space that don't touch each other, nor do their respective borders.

To elaborate on the textbook problem's steps, for any two sets A and B to be separated, the conditions \( \overline{A} \cap B = \emptyset \) and \( A \cap \overline{B} = \emptyset \) must both be met. We use the terms closure (indicated by \( \overline{A} \) or \( \overline{B} \)) to represent all points in a set, plus any 'edge' or limit points that may not be in the set itself but are so close to the set they effectively form part of its boundary.

Parts (a), (b), and (c) of the textbook exercise provide various scenarios where sets can be shown to be separated, whether they are open, closed, or defined by a particular distance from a point. Understanding this principle is essential for analyzing the structure and properties of spaces.

Uncountable Sets

In mathematics, uncountable sets are larger than any finite or countably infinite set – meaning their elements cannot be listed in a sequence. The concept of countability is linked to the ability to match a set's members with natural numbers (1, 2, 3, ...), forming a bijection.

To tie this into the textbook exercise, we encounter the proof in part (d) which demonstrates that a connected metric space with at least two points must be uncountable. Here, the idea is that if the space were countable, it would induce a certain property on continuous functions that is not possible due to separated sets existing within the space.

The proof uses a clever contradiction – it assumes there's such a bijection to show that it should be able to map the separation of two sets (as shown in part c), which is impossible. Therefore, it concludes the space must be uncountable. These sets are crucial to many areas in mathematics and are a key to understanding the structure of real numbers and functions that map between spaces.