Problem 4 - Analysis Now

Question

Show that \(\partial(Y \cup Z) \subset \partial Y \cup \partial Z\) for any two subsets \(Y\) and \(Z\) of a topological space \((X, \tau)\).

Step-by-Step Solution

Answer

\( \partial(Y \cup Z) \subseteq \partial Y \cup \partial Z \) is shown using closure properties and De Morgan's laws.

Step 1: Understanding the Boundary

The boundary of a subset \( A \) in a topological space \( (X, \tau) \) is defined as \( \partial A = \overline{A} \cap \overline{X \setminus A} \), where \( \overline{A} \) and \( \overline{X \setminus A} \) are the closures of \( A \) and the complement of \( A \) in \( X \), respectively. We will use this definition to approach the proof.

Step 2: Expressing \( \partial(Y \cup Z) \)

To express \( \partial(Y \cup Z) \), we use the definition: \[ \partial(Y \cup Z) = \overline{Y \cup Z} \cap \overline{X \setminus (Y \cup Z)}. \] By De Morgan's laws, \( X \setminus (Y \cup Z) = (X \setminus Y) \cap (X \setminus Z) \). Therefore, \[ \partial(Y \cup Z) = \overline{Y \cup Z} \cap \overline{(X \setminus Y) \cap (X \setminus Z)}. \]

Step 3: Using Properties of Closure

The closure of a union satisfies \( \overline{Y \cup Z} = \overline{Y} \cup \overline{Z} \), and the closure distributes over intersections such that \( \overline{(X \setminus Y) \cap (X \setminus Z)} \subseteq \overline{X \setminus Y} \cap \overline{X \setminus Z} \). Use these to rewrite the boundary: \[ \partial(Y \cup Z) \subseteq (\overline{Y} \cup \overline{Z}) \cap (\overline{X \setminus Y} \cap \overline{X \setminus Z}). \]

Step 4: Breaking Down the Intersection

Further express the intersection: \[ (\overline{Y} \cup \overline{Z}) \cap (\overline{X \setminus Y} \cap \overline{X \setminus Z}) = (\overline{Y} \cap \overline{X \setminus Y}) \cup (\overline{Z} \cap \overline{X \setminus Z}). \] Here, we can notice that \( \overline{Y} \cap \overline{X \setminus Y} = \partial Y \) and \( \overline{Z} \cap \overline{X \setminus Z} = \partial Z \). This means \( (\overline{Y} \cup \overline{Z}) \cap (\overline{X \setminus Y} \cap \overline{X \setminus Z}) \subseteq \partial Y \cup \partial Z \).

Step 5: Conclusion of Inclusion

Combine the results: \( \partial(Y \cup Z) \) is shown to be a subset of a union of intersections, which further leads to \( \partial(Y \cup Z) \subseteq \partial Y \cup \partial Z \). This completes the proof that the boundary of a union is contained within the union of boundaries.

Key Concepts

Boundary of a Set

In topology, when we talk about the 'boundary' of a set, we're referring to the points that define the edge of that set within a given space. The boundary of a subset \( A \) in a topological space \( (X, \tau) \) is symbolized by \( \partial A \). This can be mathematically defined as:

• \( \partial A = \overline{A} \cap \overline{X \setminus A} \)

Here:

• \( \overline{A} \) is the closure of set \( A \).
• \( \overline{X \setminus A} \) is the closure of the complement of \( A \) in \( X \).

What does this mean practically? Picture a shape on a piece of paper.

• The boundary is the line that separates the inside from the outside.
• Inside points are part of the set without being on the edge.
• Outside points are beyond the boundary.

This boundary doesn't strictly belong to the set or its complement, but provides a crucial transition zone. Understanding boundaries helps us recognize where changes occur between different regions in topological spaces.

Closure in Topology

The concept of the 'closure' of a set in topology extends our understanding of what it means for a set to contain all its limit points. When considering a subset \( A \) of a topological space \( X \), the closure, written as \( \overline{A} \), comprises all points in \( A \) plus any limit points that might lie on the edge of \( A \). Simply put, the closure \( \overline{A} \) is the 'filled-in' version of \( A \), including points that are arbitrarily close.
In mathematical terms:

• The closure \( \overline{A} \) consists of the set \( A \) union with \( A' \) (the derived set of points approachable as limit points of sequences within \( A \)).

Properties of the closure include:

• Being the smallest closed set containing \( A \).
• Being closed itself in the topological sense.
• Including the boundary of \( A \), which is \( \partial A = \overline{A} \cap \overline{X \setminus A} \).

By encompassing all these features, the closure of \( A \) ensures that we comprehend the full extent of set containment within a topological structure, highlighting its connections to the surrounding space.

De Morgan's Laws

De Morgan's Laws are vital logical principles that link the operations of union and intersection with complementation. They express how mathematical operations interact when involving sets and their complements. These laws are essentially the rules of thumb for working with Boolean algebra and set theory, and in the context of topology, they become critical in understanding how various topological operations distribute over one another.The two primary laws are:

• \( \overline{Y \cup Z} = \overline{Y} \cap \overline{Z} \)
• \( \overline{Y \cap Z} = \overline{Y} \cup \overline{Z} \)

Here, \( \overline{Y \cup Z} \) exemplifies the principle of distributing complements across a union and changing it into an intersection (and vice-versa), illustrating that:

• The complement of a union is the intersection of complements.
• The complement of an intersection is the union of complements.

In topology, applying De Morgan’s laws allows us to rewrite and simplify expressions for boundaries, closures, and other set operations, ensuring that complex operations become more manageable. For example, when analyzing \( \partial(Y \cup Z) \), De Morgan's laws can help refactor the complement of the union into a more tractable form, pivotal in proving relationships like \( \partial(Y \cup Z) \subseteq \partial Y \cup \partial Z \).