Question

(a) Show that the intersection of finitely many open sets is open. (b) Give an example showing that the intersection of infinitely many open sets may fail to be open.

Short answer

Question: Prove that the intersection of a finite number of open sets is open, and provide an example where the intersection of infinitely many open sets is not open. Answer: (a) To prove that the intersection of a finite number of open sets is open, let's consider a collection of finitely many open sets, say {U1, U2, ..., Un}. For any point x in the intersection, there exists an open ball B(x, ri) centered at x with radius ri such that the open ball is contained entirely within Ui for each i. If we consider the smallest of these radii, r = min{r1, r2, ..., rn}, the open ball B(x, r) is contained in all the Ui's, and thus in their intersection. Therefore, the intersection of finitely many open sets is open. (b) An example where the intersection of infinitely many open sets is not open is the collection of open intervals Un = (-1/n, 1/n) for n = 1, 2, 3, .... The intersection of these intervals is {0}, which is a singleton set containing only one point and is not open in the real line.

Step-by-step solution

Prove the intersection of finitely many open sets is open

To prove this, let's consider a collection of finitely many open sets, say \(\{U_1, U_2, \dots, U_n\}\). We need to show that the intersection of these open sets is open too. Let \(x\) be any point in the intersection, i.e., \(x \in \cap_{i=1}^n U_i\). Since \(x\) is inside all the open sets, we know that for each \(U_i\), there exists an open ball \(B(x, r_i)\) centered at \(x\) with radius \(r_i\) such that the open ball is contained entirely within \(U_i\). Now, let's consider the smallest of these radii, say \(r = \min\{r_1, r_2, \dots, r_n\}\). Then, the open ball \(B(x, r)\) is contained in all the \(U_i\)'s, as the smallest radius ensures that it is entirely within each \(U_i\). Thus, \(B(x, r) \subseteq \cap_{i=1}^n U_i\). Since this is true for any arbitrary point x in the intersection, we conclude that the intersection of finitely many open sets is open.

Provide an example where the intersection of infinitely many open sets is not open

Let's consider the following collection of open intervals in the real line: $$U_n = \left(\frac{-1}{n}, \frac{1}{n}\right), \, \text{for } n = 1, 2, 3, \dots$$ Note that each of the intervals \(U_n\) is an open set. Now let's find their intersection: $$\bigcap_{n=1}^{\infty} U_n = \left\{x \in \mathbb{R} \mid x \in U_n, \, \forall n \in \mathbb{N}\right\}$$ We can see that only \(x = 0\) satisfies the condition that \(x \in U_n\) for all \(n \in \mathbb{N}\). Thus, the intersection is \(\bigcap_{n=1}^{\infty} U_n = \{0\}\), which is a singleton set containing only one point. Singleton sets are not open in the real line, since there's no open interval which is a subset of this set and contains zero. Hence, part (b) shows that the intersection of infinitely many open sets may not be open.

Key concepts

Topology

Topology is a fundamental area of mathematics concerned with the properties of space that are preserved under continuous transformations. Topology examines concepts like continuity, compactness, and connectedness. One crucial aspect of topology involves understanding different kinds of sets, such as open and closed sets. These are critical in defining the structure and nature of topological spaces. For example, in the realm of topology, the concept of an open set is pivotal. An open set can be visualized as a "flexible" kind of space, where any point inside is surrounded by other points of the same set. This characteristic is intrinsic to understanding how spaces behave under continuous deformations, which do not tear or glue points together, such as stretching or bending. Topology's broad application spans various fields, including

• Geometry
• Analysis
• Various branches of modern physics

where the abstract notion of spaces helps describe and examine multidimensional shapes and structures. Understanding topology allows students to get a grip on how mathematics can describe spatial phenomena in diverse contexts.

Open Sets

An open set is a foundational idea in topology and real analysis that defines a kind of space or subset of a given space. Open sets are characterized by the fact that for any point within the set, there exists a neighborhood around it that is entirely contained within the set itself. This means that if you are standing at a point inside an open set, you can always find a "safe zone" or "breathing space" entirely within that set, free of any "edges." In the context of the exercise, showing that the intersection of finitely many open sets is open requires understanding that each point within this intersection can similarly have a neighborhood contained entirely within the intersection. For example, the intersection of three open sets, each represented as circles on a flat surface, where any point belonging to all three has an area around it wholly inside all three. However, when you consider infinitely many open sets, like shrinking intervals along the real line towards a single point, the intuitive "open" nature fades. The singleton emerges, losing the "zone" of openness, illustrating that intersections can sometimes dramatically change the nature of a set.

Real Analysis

Real analysis focuses on the behavior of real numbers, sequences, series, and real-valued functions. It is a rigorous branch of mathematics that seeks to understand detailed and precise processes involving real numbers, which includes discussions involving limits, continuity, derivatives, and integration. Key to real analysis is the study of open and closed sets within the real number line. These sets define important criteria for determining the continuity of functions and convergence of sequences. Open sets form the basis of the topology on the real line, helping to define which functions are continuous, which sequences converge, and how measures are calculated. In the exercise, the importance of real analysis becomes apparent when illustrating how the intersection of infinitely many open intervals results in a set that is no longer open. Here, understanding real numbers' behavior helps reveal why a singleton, like {0}, lacks the extending "safety zone" characteristic of open sets. Real analysis, therefore, equips students with a profound toolkit to rigorously assess and interpret the continuous phenomena often observed in both pure and applied mathematics. Engaging with real analysis provides insights into everything from solving differential equations to optimizing functions, making it a cornerstone of advanced mathematical study.