Question

A topological space \(X\) is called disconnected if there exist nonempty open sets \(U, V\) such that \(U \cap V=\varnothing\) and \(U \cup V=X\); otherwise \(X\) is connected. When we speak of connected or disconnected subsets of \(X\), we refer to the relative topology on them. a. \(X\) is connected iff \(\varnothing\) and \(X\) are the only subsets of \(X\) that are both open and closed. b. If \(\left\\{E_{\alpha}\right\\}_{\alpha \in A}\) is a collection of connected subsets of \(X\) such that \(\bigcap_{\alpha \in A} E_{\alpha} \neq\) \(\varnothing\), then \(\bigcup_{\alpha \in A} E_{\alpha}\) is connected. c. If \(A \subset X\) is connected, then \(\bar{A}\) is connected. d. Every point \(x \in X\) is contained in a unique maximal connected subset of \(X\), and this subset is closed. (It is called the connected component of \(x\).)

Short answer

a. Connected iff only \(\varnothing\) and \(X\) are clopen. b. Union of overlapping connected subsets is connected. c. Closure of a connected set is connected. d. Each point is in a unique maximal connected closed subset.

Step-by-step solution

Defining the Problem

To solve this exercise, we need to understand the concepts of connected and disconnected topological spaces and subsets. A space \(X\) is connected if there are no nonempty open sets whose union covers the space and whose intersection is empty.

Part a: Prove Condition for Connectedness

We need to show that \(X\) is connected if and only if the only subsets of \(X\) that are both open and closed (clopen) are \(\varnothing\) and \(X\). If \(E\) is both open and closed, then \(E\cup (X\setminus E) = X\) where both \(E\) and \(X\setminus E\) are open, contradicting connectivity if \(Eeq \varnothing\) and \(Eeq X\). Conversely, if only \(\varnothing\) and \(X\) are clopen, \(X\) must be connected.

Part b: Show Union of Overlapping Connected Sets is Connected

We need to prove that if we have a collection \(\{E_\alpha\}_{\alpha \in A}\) of connected subsets with a non-empty intersection, their union \(\bigcup_{\alpha \in A} E_\alpha\) is connected. Since each \(E_\alpha\) is connected and they intersect non-trivially, any separation of one would separate the whole union, which results in a contradiction. Thus, the union must be connected.

Part c: Show Closure of a Connected Set is Connected

Prove that if \(A\subset X\) is connected, then \(\bar{A}\) (its closure) is also connected. Assume for contradiction that \(\bar{A}\) can be decomposed into two disjoint open sets \(U\) and \(V\), such that \(U \cup V = \bar{A}\). Since \(A\subset \bar{A}\), \(A\subseteq (U \cup V)\), but \(A\) being connected means this cannot happen unless one is empty, so \(\bar{A}\) must be connected.

Part d: Show Existence and Uniqueness of Maximal Connected Subset

To prove that every point \(x \in X\) belongs to a unique maximal connected subset called its connected component, consider the union of all connected subsets containing \(x\). This union is connected by part b, and it contains \(x\). If another connected set contains \(x\), it must overlap with this union, hence is a subset of it, proving maximality. This component is closed because its complement is open and does not intersect with it.

Key concepts

Disconnected Spaces

A topological space is termed "disconnected" when we can find two non-empty open sets, say \(U\) and \(V\), that meet these conditions:

• The intersection of \(U\) and \(V\) is empty, denoted by \(U \cap V = \emptyset\).
• Their union covers the entire space, meaning \(U \cup V = X\).

In simple terms, a disconnected space can be split into two distinct parts that do not overlap, mimicking the idea of splitting something into two separate pieces.
If a space cannot be split in such a way, it is labeled as "connected." This concept of splitting carries over to subsets of a given space. When subsets have their own topology drawn from the larger space, they can also be examined for their connectivity in this relative sense.

Connected Subsets

Within a topological space, subsets can be connected or disconnected. Understanding when a subset is connected is crucial to understanding the structure of the space itself. A subset \(A\) of space \(X\) is called "connected" if no separation into two disjoint open sets exists. This means you can't split \(A\) into two open sets without one of them being empty. Moreover, collections of connected subsets having non-empty intersections are significant. If each subset in such a collection is connected, then their union will be connected as well, provided that the subsets have a common intersection. This non-empty intersection ensures cohesiveness, preventing a split that would otherwise separate them.

Clopen Sets

The concept of clopen sets—sets that are both open and closed—is pivotal in understanding connected spaces. In a connected space \(X\), the only clopen sets are \(\emptyset\) and \(X\) itself.
Here's why:

• If there exists a non-trivial clopen set \(E\) within \(X\), we can express \(X\) as the union of \(E\) and its complement \(X \setminus E\), both open. This directly conflicts with the definition of a connected space.
• Such a non-trivial \(E\) suggests a kind of disconnection, because it partitions \(X\) into two separate areas.

Consequently, the non-existence of any clopen sets except \(\emptyset\) and \(X\) implies that the space is connected. In topology, this idea of clopen sets serves as a litmus test for connectivity.

Closure of Connected Sets

Another fascinating idea in topology is the behavior of connected sets when closed. Consider a connected subset \(A\) within a space \(X\). The closure of \(A\), denoted \(\bar{A}\), is also connected. Here's why:

• Imagine if \(\bar{A}\) could be split into two disjoint open sets, \(U\) and \(V\), such that \(U \cup V = \bar{A}\).
• Because \(A\) is within \(\bar{A}\), any supposed division in \(\bar{A}\) would also apply to \(A\).
• Since \(A\) cannot be split due to its connected nature, it follows that \(\bar{A}\) can't be split either unless one of the sets is empty.

This property ensures continuity beyond the boundaries of the set, emphasizing the robustness of connectedness through closure. It illustrates that connectedness is preserved not only within a set but also extends to its boundary.