Problem 18 - Advanced Calculus

Question

(a) Show that any two disjoint nonempty open sets are mutually separated. (b) Show that any two disjoint nonempty closed sets are mutually separated.

Step-by-Step Solution

Answer

Any two disjoint nonempty open sets or closed sets are mutually separated because no point in one set can be a limit point of the other set.

Step 1: Prove that disjoint nonempty open sets are mutually separated

Let \( U \) and \( V \) be two disjoint nonempty open sets. For \( U \) and \( V \) to be mutually separated, every point in \( U \) is an interior point of \( U \) and is, therefore, not a limit point of \( V \) since \( U \) and \( V \) are disjoint. Similarly, every point in \( V \) is not a limit point of \( U \). Hence, \( U \) and \( V \) are mutually separated.

Step 2: Prove that disjoint nonempty closed sets are mutually separated

Now, let's let \( A \) and \( B \) be two disjoint nonempty closed sets. Since \( A \) is closed, it contains all its limit points. However, since \( A \) and \( B \) are disjoint, all of \( A \)'s limit points are not in \( B \). Therefore, \( B \) does not contain any limit point from \( A \). Same reasoning applies to set \( B \). Therefore, \( A \) and \( B \) are mutually separated.

Key Concepts

Open Sets

In topology, an open set is a fundamental concept. Think of an open set as a collection of points, each having some space around them within the set. The idea is that no boundary point belongs to the set, much like the interior of a circle without its border.

Here are a few key characteristics of open sets:

• Every point in an open set is an interior point. This means that for any point, you can find a small region around it that also lies entirely within the set.
• In a topological space, both the entire space and the empty set are open.
• The union of any collection of open sets is also open.

In the context of the exercise, when two open sets are disjoint (having no elements in common), it means each point in one set is isolated from the other, thus they are mutually separated.

Closed Sets

Closed sets in topology are those that include all their boundary points. To help visualize this, imagine the entirety of a circle, including its border.

Key features of closed sets include:

• A closed set contains all its limit points. If you have a sequence of points inside the set converging to a limit point, that point is also in the set.
• The intersection of any collection of closed sets is closed.
• The complement of an open set is a closed set, and vice versa.

For the given exercise, when two closed sets are disjoint, none of the boundary points of one set are in the other. This ensures that the sets are mutually separated, reinforcing their distinct nature.

Limit Points

Limit points are intriguing elements in topology. A limit point of a set is a point such that every neighborhood around it contains another point from the set. It's like a point constantly approached by other points in the set.

Understanding limit points:

• If you have a sequence that converges to a point, that point is a limit point.
• A point can be a limit point without actually being part of the set.
• In closed sets, all limit points of the set are included in the set itself.

In terms of disjoint sets, any limit point of one set cannot be in the other, ensuring they remain distinctly separate.

Disjoint Sets

Disjoint sets are sets that do not share any common elements. They are like two groups with completely separate members.

Characteristics of disjoint sets:

• They have an intersection that is empty. In other words, there is no overlap between them.
• Disjoint sets are critical in understanding concepts like separation in topology because they emphasize distinct boundaries between sets.

In the context of the exercise, showing that open or closed sets are disjoint involves proving that no points can connect them, reinforcing their independent nature.