Question

Suppose that no two of the sets \(A_{i}(i \in I)\) are disjoint. Prove that if all \(A_{i}\) are connected, so is \(A=\bigcup_{i \in I} A_{i}\) [Hint: If not, let \(A=P \cup Q(P, Q\) as in Definition 3). Let \(P_{i}=A_{i} \cap P\) and \(Q_{i}=A_{i} \cap Q,\) so \(A_{i}=P_{i} \cup Q_{i}, i \in I\) That is, onto a two-point set \\{0\\}\(\cup\\{1\\}\). \S10. Arcs and Curves. Connected Sets At least one of the \(P_{i}, Q_{i}\) must be \(\emptyset\) (why?); say, \(Q_{j}=\emptyset\) for some \(j \in I\). Then \((\forall i) Q_{i}=\emptyset,\) for \(Q_{i} \neq \emptyset\) implies \(P_{i}=\emptyset,\) whence $$ A_{i}=Q_{i} \subseteq Q \Longrightarrow A_{i} \cap A_{j}=\emptyset\left(\text { since } A_{j} \subseteq P\right) $$ contrary to our assumption. Deduce that \(Q=\bigcup_{i} Q_{i}=\emptyset\). (Contradiction!)]

Short answer

\(A\) must be connected by contradiction.

Step-by-step solution

Identify the Problem

We need to prove that if all sets \(A_i\) are connected and no two are disjoint, then the union \(A = \bigcup_{i \in I} A_i\) is connected. We are given a hypothesis for contradiction.

Assume for Contradiction

Assume the contrary: that \(A\) is not connected. Then by definition, \(A = P \cup Q\) where both \(P\) and \(Q\) are non-empty and disjoint open sets.

Define Subsets

Define \(P_i = A_i \cap P\) and \(Q_i = A_i \cap Q\) for each \(i \in I\). Thus, \(A_i = P_i \cup Q_i\).

Apply Connectedness

Given \(A_i\) is connected, either \(P_i\) or \(Q_i\) must be empty for each \(i\). Assume \(Q_j = \emptyset\) for some \(j\).

Show Consequences of Assumption

If \(Q_j = \emptyset\), for all \(i, Q_i = \emptyset\) since \(Q_i eq \emptyset\) would imply \(P_i = \emptyset\), leading to \(A_i \subseteq Q\). This makes \(A_i \cap A_j = \emptyset\), against our given condition that no two sets are disjoint.

Conclude with Contradiction

The assumption that \(Q\) is non-empty results in a contradiction because \(\bigcup_i Q_i = \emptyset\). Thus, \(A\) must be connected.

Key concepts

Union of Sets

When dealing with sets, the **union of sets** is a fundamental concept you will encounter frequently. The union of a collection of sets is essentially a new set that contains all elements that are in any of the original sets.

• For example, if you have two sets, say set A = \{1, 2, 3\} and set B = \{3, 4, 5\}, the union, denoted as \(A \cup B\), would be \{1, 2, 3, 4, 5\}.

In the context of connected sets, if each individual set in a union is connected, and no two of them are disjoint, this relationship can have implications for the connectedness of the union itself.
If all sets \(A_i\) are connected, and no pair of these sets is disjoint, it suggests a kind of overlap or glue that holds the entire union together continuously, contributing to the connectedness of the union itself.

Disjoint Sets

**Disjoint sets** are those which have no elements in common. Mathematically, if set A and set B are disjoint, then \(A \cap B = \emptyset\).

• For instance, consider two sets C = \{1, 2\} and D = \{3, 4\}. These are disjoint because they share no elements.

In the given problem, we are told that none of the sets \(A_i\) are disjoint. This means each \(A_i\) has at least one element in common with every other \(A_j\).
This is a key part of proving the connectedness of the union, as it implies there's some intersection linking the sets together, making it impossible to split the union into two non-touching, separate parts. This conceptual glue ensures that the whole structure remains a single connected entity.

Proof by Contradiction

A **proof by contradiction** is a logical method where you start by assuming the opposite of what you want to prove. If this assumption leads to a contradiction, the original statement must be true.

• In our exercise, we assume the opposite of what we want to prove, namely that the union \(A = P \cup Q\) is not connected, meaning it can be split into two disjoint non-empty sets \(P\) and \(Q\).

As a result, we find a contradiction by showing that no such division is possible without violating another given condition. In this case, assuming \(Q\) is non-empty eventually leads to the conclusion \(Q = \emptyset\), which is a contradiction, thus proving that our assumption was false, and hence, \(A\) is indeed connected.

Topology

**Topology** is an area in mathematics focused on the properties of space that are preserved under continuous transformations, such as stretching or twisting, but not tearing or gluing. One key aspect of topology is the study of connectedness in sets.

• A set is considered connected in topology if it cannot be divided into two or more disjoint open subsets.

The notion of open sets is crucial here since it helps in defining and understanding how sets remain "stuck together" even under continuous deformations.
In our context, connectedness plays a fundamental role in determining if the union of several sets, like \(igcup_{i} A_i\), holds together as one piece. By understanding these topological properties, we're able to conclude whether the union formed from these sets remains connected, proving invaluable in higher-level mathematical analysis.