Question

Let \(X\) be a metric space in which every infinite subset has a limit point. Prove that \(X\) is separable. Hint: Fix \(\delta>0\), and pick \(x_{1} \in X .\) Having chosen \(x_{1}, \ldots, x_{j} \in X\), choose \(x_{j+1} \in X\), if possible, so that \(d\left(x_{i}, x_{j+1}\right) \geq \delta\) for \(i=1, \ldots, j\). Show that this process must stop after a finite number of steps, and that \(X\) can therefore be covered by finitely many neighborhoods of radius \(\delta\). Take \(\delta=1 / n(n=1,2,3, \ldots)\), and consider the centers of the corresponding neighborhoods.

Short answer

In the given metric space \(X\), the procedure of picking a point \(x_1\), and then iteratively picking points with a distance of at least \(\delta\) away from the previous points, must stop after a finite number of steps. This is because if the process could continue indefinitely, we would have an infinite sequence of points with a limit point, contradicting the fact that no two points in the sequence are within \(\delta\) distance from each other. By repeating this process for \(\delta = \frac{1}{n}\) for each positive integer \(n\), we construct a countable dense subset \(D\) in \(X\), making \(X\) separable.

Step-by-step solution

Fix a value of \(\delta > 0\) and choose the first point

Choose an arbitrary \(\delta > 0\), and pick any point \(x_1 \in X\). Now we will iteratively choose more points in \(X\).

Iteratively choose new points in \(X\)

For every \(j \in \mathbb{N}\), suppose we have already chosen \(x_1, \ldots, x_j \in X\). We will now try to choose a new point \(x_{j+1} \in X\) such that \(d(x_i, x_{j+1}) \geq \delta\) for all \(i = 1, \ldots, j\). If we can't find such a point, the process stops.

Prove the process must stop after a finite number of steps

Assume, for the sake of contradiction, that the process never stops, and we can choose an infinite sequence of points \(x_1, x_2, x_3, \ldots\) with \(d(x_i, x_j) \geq \delta\) for all \(i \neq j\). Then, this infinite sequence forms an infinite subset of \(X\). But since every infinite subset of \(X\) has a limit point, there exists a limit point \(x \in X\) of this sequence. That means that for every \(\varepsilon > 0\) there exists \(i \in \mathbb{N}\) such that \(d(x, x_i) < \varepsilon\). Now, let \(\varepsilon = \frac{\delta}{2}\). This means that there exists \(i \in \mathbb{N}\) such that \(d(x, x_i) < \frac{\delta}{2}\). Since \(x\) is a limit point of the sequence, there must also exist another index \(j \neq i\) such that \(d(x, x_j) < \frac{\delta}{2}\). Using the triangle inequality, we get: \(d(x_i, x_j) \leq d(x_i, x) + d(x, x_j) < \frac{\delta}{2} + \frac{\delta}{2} = \delta\). But this contradicts our assumption that \(d(x_i, x_j) \geq \delta\) for all \(i \neq j\). Therefore, our initial assumption that the process can continue indefinitely is false, and the process must stop after a finite number of steps.

Cover \(X\) with finitely many \(\delta\)-radius neighborhoods

Let \(x_1, x_2, \ldots, x_n\) be the finitely many points we obtain from the process. Now for each \(i = 1, \ldots, n\), let \(N_i = \{y \in X \mid d(y, x_i) < \delta\}\), which is the open neighborhood centered at \(x_i\) with radius \(\delta\). Since every point in \(X\) is at least distance \(\delta\) away from some \(x_i\), with this process we will cover \(X\) with finitely many open neighborhoods.

Repeat the process for \(\delta = \frac{1}{n}\), and construct a countable dense subset

Now, we repeat the above process for \(\delta = \frac{1}{n}\), where \(n = 1, 2, 3, \ldots\). We obtain a sequence of finite sets of points \(A_n = \{x^{(n)}_1, \ldots, x^{(n)}_{k_n}\}\) such that \(X\) can be covered by the corresponding open neighborhoods with radii \(\frac{1}{n}\) for each \(n \in \mathbb{N}\). Then, we form a set \(D = \bigcup_{n=1}^{\infty} A_n\). Since the union of countably many finite sets is countable, \(D\) is a countable set. Finally, we need to show that \(D\) is dense in \(X\). Let \(y \in X\) be an arbitrary point, and let \(\varepsilon > 0\) be given. Choose an integer \(N > \frac{2}{\varepsilon}\), and consider the process with \(\delta = \frac{1}{N}\). Since \(X\) can be covered by neighborhoods with radius \(\frac{1}{N}\), there must be a point \(x \in A_N\) such that \(d(y, x) < \frac{1}{N}\). But then, since \(\frac{1}{N} < \frac{\varepsilon}{2}\), we have \(d(y, x) < \varepsilon\). Since \(y\) and \(\varepsilon\) were arbitrary, this shows that \(D\) is dense in \(X\), and therefore \(X\) is separable.

Key concepts

Limit Point

A limit point, or accumulation point, is a fundamental concept in the field of mathematical analysis, especially in the context of metric spaces. In simple terms, a limit point of a set in a metric space is a point that can be 'approached' by other points of the set, in the sense that any arbitrarily small region around the limit point will contain some other points of the set.

For a formal definition, consider a metric space \(X, d\), where \(d\) is the distance function. A point \(x \in X\) is called a limit point of a subset \(A \subseteq X\) if for every radius \(r > 0\), the open ball \(B(x, r)\), which includes all points in \(X\) within distance \(r\) from \(x\), contains at least one point of \(A\) other than \(x\) itself.

This concept is used to identify the 'boundary' behavior of a set and plays a crucial role in defining continuity, convergence, and some of the most exciting results in analysis, such as the Bolzano-Weierstrass theorem.

Metric Space Properties

Metric spaces are equipped with a set of properties defined through a function called a metric, which essentially is a way to talk about distances in the space. The most significant properties of metric spaces are that they provide a concrete method to measure the distance between any two points, and they satisfy certain conditions that make them intuitive and useable for mathematical analysis.

Let's walk through these metric space properties:

• Positivity: The distance between two points is always a non-negative number (\(d(x, y) \geq 0\)).
• Identity of Indiscernibles: The distance is zero if and only if the two points are the same (\(d(x, y) = 0 \Leftrightarrow x = y\)).
• Symmetry: The distance is the same regardless of the order of points (\(d(x, y) = d(y, x)\)).
• Triangle Inequality: The distance from point \(x\) to point \(z\) is less than or equal to the sum of the distance from \(x\) to \(y\) and from \(y\) to \(z\) (\(d(x, z) \leq d(x, y) + d(y, z)\)).

The properties of metric spaces are essential for defining open and closed sets, continuity, and for the analysis of functions and sequences within these spaces.

Dense Subset

In the realm of topology and analysis, a dense subset carries immense importance due to its role in approximation. A subset \(D\) of a metric space \(X\) is said to be dense in \(X\) if every point in \(X\) is either in \(D\) or is a limit point of \(D\). Alternatively, \(D\) is dense in \(X\) if for every point \(x \in X\) and every \(\varepsilon > 0\), there is a point \(d \in D\) such that \(d(x, d) < \varepsilon\).

This property means that within any part of the space, no matter how small, you can find points from the dense subset. Such subsets are crucial for they allow us to 'approximate' any point in the space with points from the subset. For instance, the rational numbers form a dense subset of the real numbers because between any two real numbers, there are infinitely many rational numbers.

Mathematical Analysis

Mathematical analysis is a branch of mathematics dealing with limits and related theories, like differentiation, integration, measure, infinite series, and analytic functions. These theories are usually studied in the context of real numbers, complex numbers, and more abstract spaces such as metric spaces.

In analysis, one explores and proves various properties of functions, sequences, sets, and more. This discipline lays the groundwork for understanding and formalizing the concepts of convergence, continuity, and compactness, which are used not only in pure mathematics but also form the foundation of various applied fields including physics, engineering, and economics.

Analysis is the theoretical backbone of calculus, and it involves rigorous proofs to ensure that the intuitive notions of 'smoothness', 'limiting behavior', and 'infinitesimal change', which are taken for granted in calculus, hold under scrutiny. It also provides tools for estimating the size of errors in calculations, known as error analysis – a critical aspect of numerical analysis.