Question

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

Short answer

Provide a reasoning for your answer. Yes, the union of finitely many closed sets is always closed. This is because the complement of the union of finitely many closed sets is the intersection of their complements, which are open sets. Since the intersection of finitely many open sets is open (by the definition of a topology), the complement of the union is an open set, which implies that the union of finitely many closed sets is a closed set.

Step-by-step solution

(Part A: Proving the union of finitely many closed sets is closed)

Suppose we have finitely many closed sets \({A_1, A_2, ..., A_n}\). To prove that their union is closed, we need to show that the complement of the union is an open set. We have: $$(A_1 \cup A_2 \cup ... \cup A_n)^C = A_1^C \cap A_2^C \cap ... \cap A_n^C.$$ Since each \(A_i\) is closed, their complements \(A_i^C\) are open sets. The intersection of finitely many open sets is open (by the definition of a topology). Therefore, the complement of the union is an open set, which implies that the union \((A_1 \cup A_2 \cup ... \cup A_n)\) of the closed sets is a closed set.

(Part B: Example of an infinite union of closed sets not being closed)

Let's consider the set of all intervals \(A_n = [\frac{1}{n}, 1]\) for \(n \in \mathbb{N}\). In this case, \(A_n\) represents an infinite collection of closed sets. Now, let's find the union of all these closed sets: $$A = \bigcup_{n=1}^{\infty} A_n = \bigcup_{n=1}^{\infty} \left[\frac{1}{n}, 1\right] = (0, 1].$$ We can see that the value \(0\) is not included in any of the intervals \(A_n\), but for any value \(\epsilon > 0\), we can always find an \(n \in \mathbb{N}\) such that \(1/n < \epsilon\), which means that the interval contains the number \(1/n\) and is part of the union. In other words, no matter how small the interval around \(0\) is, there exist numbers from the range \((0, \epsilon)\) that are included in the union. Thus, the union of all these intervals is \((0, 1]\). The interval \((0, 1]\) is not closed, since its complement \((-\infty, 0] \cup (1, \infty)\) is not an open set. This example shows that the union of infinitely many closed sets might not be closed.

Key concepts

Closed Sets

In topology, a closed set is complementary to an open set within a given space. Closed sets are an essential part of understanding topological structures because their properties can tell us a lot about the space they're in. Here are some key characteristics of closed sets:

• Complements: The complement of a closed set is an open set. In other words, if you take away a closed set from the entire space, what you have left is an open set.
• Containing Limit Points: Closed sets contain all their limit points. This means if you take any point that "almost" belongs to the closed set (approaching but not necessarily included), this point is actually within the closed set.
• Union and Intersection: The union of a finite number of closed sets results in a closed set. Also, the intersection of any collection of closed sets is also closed.

Understanding these characteristics helps in visualizing how closed sets behave in different spaces. They form a counterpart to open sets in the comprehension of topological spaces.

Union of Sets

The concept of a union in set theory involves combining elements of multiple sets. When considering topology, how sets unite—especially closed sets—affects their comprehensive nature. Consider these aspects of union:

• Finite Union: When you unite a finite number of closed sets, the result is a closed set. This relies on the fact that the complement of a finite union of closed sets is an intersection of open sets, which is open.
• Infinite Union: The union of infinitely many closed sets doesn't necessarily result in a closed set. This is crucial because it highlights a fundamental difference in behavior compared to finite unions.

For instance, the infinite union of the intervals \( \left[ \frac{1}{n}, 1 \right] \) for \( n \in \mathbb{N} \) results in the interval \( (0, 1] \), which is not closed. Such examples remind us how topological properties change with infinite operations.

Open Sets

Open sets are the foundational building blocks in topology. While closed sets offer one perspective, open sets allow us to examine the "open-ness" or "accessibility" within a space. Here's what makes open sets vital:

• Defining Characteristic: A set is open if, for every point within the set, there is a neighborhood completely contained in the set. This aligns with our intuitive understanding of openness.
• Complements: The complement of an open set is closed, meaning that there's a clear duality between open and closed sets.
• Familiar Examples: Common examples include open intervals in real numbers, such as \( (a, b) \), where all the points between a and b, excluding the endpoints, comprise the open set.

In topology, knowing whether a set is open or closed helps in understanding the "shape" or structure of spaces, determining continuous functions, and exploring limits and convergence. Open sets thus provide a framework for what it means to "move freely" within a topological space.