Problem 7 - Basic Analysis: Introduction to Real Analysis

Question

In any metric space, prove: a) \(E\) is closed if and only if \(\partial E \subset E\). b) \(U\) is open if and only if \(\partial U \cap U=\emptyset\).

Step-by-Step Solution

Answer

E is closed if \(\partial E \subset E\); U is open if \(\partial U \cap U = \emptyset\).

Step 1: Define the boundary of a set

The boundary of a set \(E\) in a metric space is defined as \(\partial E = \overline{E} \cap \overline{E^c}\), where \(\overline{E}\) is the closure of \(E\) and \(E^c\) is the complement of \(E\).

Step 2: Prove E is closed if and only if \( \partial E \subset E \)

For \(E\) to be closed, all its limit points must be included in \(E\). By definition, \(\partial E = \overline{E} \cap \overline{E^c}\). \(E\) is closed means \(E = \overline{E}\), so \(\partial E = E \cap \overline{E^c}\). Hence, \(\partial E \subset E\). Conversely, if \(\partial E \subset E\), then all points that would need to be added to make \(E\) closed are already in \(E\), so \(E\) is indeed closed.

Step 3: Prove U is open if and only if \(\partial U \cap U = \emptyset\)

A set \(U\) is open if \(U = \text{int}(U)\), meaning no boundary points belong to \(U\). \(\partial U = \overline{U} \cap \overline{U^c}\). If \(U\) is open, then \(U \supset \overline{U^c}\), leading to \(\partial U \cap U = \emptyset\). Conversely, if \(\partial U \cap U = \emptyset\), no boundary points of \(U\) are in \(U\), implying that \(U\) is open as all points in \(U\) have a neighborhood that lies entirely in \(U\).

Key Concepts

Boundary of a set

In the context of metric spaces, the boundary of a set plays a crucial role in understanding its properties. The boundary of a set \(E\) is denoted by \(\partial E\) and is defined using both the closure of the set and its complement. Formally, it is expressed as:

• \(\partial E = \overline{E} \cap \overline{E^c}\)

Here, \(\overline{E}\) represents the closure of the set \(E\), which includes all the limit points of \(E\). The complement of \(E\), denoted by \(E^c\), consists of all the points not in \(E\). Thus, \(\overline{E^c}\) would be the closure of the complement of \(E\). The boundary is the set of points that can be "approached" by points in \(E\) and also by points not in \(E\).
This boundary forms a kind of "edge" for the set \(E\) in the larger metric space, representing the transitional zone where the set meets its surroundings. Understanding the boundary helps determine other characteristics of the set, such as whether it's open or closed.

Open and closed sets

In metric spaces, the concepts of open and closed sets are foundational for understanding the structure and behavior of sets. A set is considered **open** if all points within it have some neighborhood entirely contained within the set. Mathematically, a set \(U\) is open if \(U = \text{int}(U)\), meaning the set is equal to its interior.
On the other hand, a **closed set** is one that contains all its limit points. This means if you were to approach the boundary from within the set, you would still be inside the set. Mathematically, a set \(E\) is closed if \(E = \overline{E}\), meaning it is equal to its closure.

Interestingly, open and closed sets can be linked to the boundary concept:

• A set \(E\) is closed if and only if its boundary \(\partial E\) is entirely contained within \(E\).
• A set \(U\) is open if its boundary \(\partial U\) has no intersection with \(U\) itself, i.e., \(\partial U \cap U = \emptyset\).

These properties of open and closed sets provide essential insights into their structure and are useful in various branches of mathematics.

Closure of a set

The closure of a set \(E\), denoted as \(\overline{E}\), is an extension that includes \(E\) itself as well as all points "touching" \(E\). In simpler terms, the closure adds all of the limit points to \(E\), filling in any "gaps" that may exist at the edge of the set. The limit points are those that can be approached as closely as desired by elements within \(E\).

The closure can be seen mathematically as:

• \(\overline{E} = E \cup \partial E\)

This expression shows that the closure involves the original set \(E\) combined with its boundary points. When these limit points are added to the set, they essentially "close" it.

The concept of closure is intertwined with open and closed sets. If a set is equal to its closure (\(E = \overline{E}\)), the set is closed. Closure operation is significant in analysis and topology because it doesn't just include what's inside but also accounts for the "borderline" cases—making its role crucial in understanding the set in its entirety.