Question

A collection \(\left\\{V_{a}\right\\}\) of open subsets of \(X\) is said to be a base for \(X\) if the following is true: For every \(x \in X\) and every open set \(G \subset X\) such that \(x \in G\), we have \(x \in V_{s} \subset G\) for some \(\alpha .\) In other words, every open set in \(X\) is the union of a subcollection of \(\left\\{V_{t}\right\\}\) Prove that every separable metric space has a countable base. Hint: Take all neighborhoods with rational radius and center in some countable dense subset of \(X\).

Short answer

Question: Prove that every separable metric space has a countable base. Answer: Given a separable metric space (X, d) with a countable dense subset A, we can construct a countable collection of open sets, denoted as B = {B(x_n, q) | x_n ∈ A, q ∈ Q}, where Q is the set of all positive rational numbers. This collection serves as a countable base for the metric space, as for every open set G ⊂ X and every x ∈ G, there exists a B(x_n, q) ∈ B such that x ∈ B(x_n, q) ⊂ G.

Step-by-step solution

Definitions and notations

First, let's recall some definitions and notations: - A metric space \((X, d)\) is separable if it has a countable dense subset. A dense subset is a subset \(A \subseteq X\) such that \(\overline{A} = X\). - A base \(\{V_a\}\) for a topological space \(X\) is a collection of open sets such that every open set in \(X\) is the union of a subcollection of \(\{V_a\}\). Now, let \((X, d)\) be a separable metric space and let \(A = \{x_n \mid n \in \mathbb{N}\}\) be a countable dense subset of \(X\). Let \(Q\) be the set of all positive rational numbers.

Construct the countable collection

Our goal is to construct a countable collection of open sets in \(X\) that will serve as a base. For this, we take the collection of open balls with centers in \(A\) and radii in \(Q\). Define the collection as follows: $$\mathcal{B} = \{B(x_n, q) \mid x_n \in A, q \in Q\}$$ where \(B(x_n, q) = \{x \in X \mid d(x, x_n) < q\}\) is an open ball centered at \(x_n\). Since \(A\) and \(Q\) are both countable sets, the collection \(\mathcal{B}\) is also countable.

Prove that the collection is a base

Now, let's prove that the collection \(\mathcal{B}\) is a base for the topological space \((X, d)\). In order to do this, we need to show that for every open set \(G \subset X\) and every \(x \in G\), there exists a \(B(x_n, q) \in \mathcal{B}\) such that \(x \in B(x_n, q) \subset G\). First, observe that every \(x \in G\) must be an interior point of \(G\). By definition of interior points, there exists an open ball \(B(x, \epsilon)\) centered at \(x\) with \(\epsilon > 0\) such that \(B(x, \epsilon) \subset G\). Now, since \(A\) is dense in \(X\), there exists an \(x_n \in A\) such that \(x_n \in B(x, \epsilon/2)\). Also, there exists a rational number \(q \in Q\) such that \(0 < q < \frac{\epsilon}{2}\). Consider the open ball \(B(x_n, q) \in \mathcal{B}\). Notice that for any \(y \in B(x_n, q)\), it holds that \(d(y, x) \le d(y, x_n) + d(x_n, x) < q + \frac{\epsilon}{2} < \epsilon\). Therefore, \(y \in B(x, \epsilon)\). This implies that \(B(x_n, q) \subset B(x, \epsilon) \subset G\). Since \(x_n \in B(x_n, q)\), we have shown that for every \(x \in G\) there exists a \(B(x_n, q) \in \mathcal{B}\) such that \(x \in B(x_n, q) \subset G\), and there exists a subcollection of \(\mathcal{B}\) whose union equals \(G\). Thus, the collection \(\mathcal{B}\) is a countable base for the metric space \((X, d)\).

Key concepts

Countable Base

In the study of topology, a countable base is a particularly convenient type of basis for describing the structure of a topological space. A base is a collection of open sets such that any open set in the space can be written as a union of sets from this collection. When the base is countable—that is, its elements can be put into one-to-one correspondence with the natural numbers—the topological space is equipped with a property that simplifies many proofs and theorems.

For example, having a countable base is essential in establishing whether certain types of convergence and continuity properties hold in a topological space. In the context of metric spaces, which are topological spaces equipped with a notion of distance, the concept of a countable base becomes even more powerful. By using open balls (sets consisting of all points within a certain distance from a center point) with rational radii, we're effectively creating a finely-tunable 'net' that can cover any open set in the space, ensuring that the topology can be described with just a countable collection of sets.

Topological Space

A topological space is a fundamental concept in topology, serving as the main object of study. It consists of a set of points along with a collection of open subsets, which define how the points relate to each other in terms of 'closeness' without necessarily defining an exact distance as in metric spaces. The open subsets must satisfy three key axioms: the entire space and the empty set must be open, any union of open sets must also be open, and the intersection of any finite number of open sets must be open.

In a topological space, many properties like continuity, convergence, and connectedness are defined purely in terms of open sets, making it a highly abstract but widely applicable concept in mathematics. These properties don't rely on a specific notion of distance, which grants topological spaces a level of generality beyond that of metric spaces, which do have such notions.

Open Subsets

In the realms of both metric and topological spaces, open subsets play a pivotal role. An open subset 'wraps around' its elements providing a cushion that excludes its border—if indeed it has one. To illustrate, consider an open interval on the number line, such as \( (a, b) \), where all the points between 'a' and 'b' are included, but 'a' and 'b' themselves are not.

When we generalize this to topological spaces, an open set doesn't require an interval structure. It simply must adhere to the topological rules: mainly, any point within an open set is surrounded by other points in the set in such a way that there is room to 'wiggle' without leaving the set. This flexibility allows for varied and exotic topologies, some of which can be quite counterintuitive compared to the familiar geometries of Euclidean spaces.

Dense Subset

A dense subset in a topological space has a characteristic of being able to 'touch' every part of the space, even if it's not intuitive at first glance. Formally, a subset 'A' of a space 'X' is called dense if every point in 'X' either belongs to 'A' or is a limit point of 'A'. This means you can get arbitrarily close to any point in 'X' by selecting points from 'A'.

The concept of a dense subset becomes especially interesting in separable spaces, those with a countable dense subset. Separability implies that the space, regardless of its size or complexity, can be thought of as being generated from a humble collection of points that can be listed in a sequence. Such spaces have strong ties to notions of 'countability', and this becomes invaluable when constructing arguments in analysis or solving problems in topology.